If you're looking to buy an old computer and you think an Acorn or three may fit the bill this very quick run-down may be of some help:

Part 1: The Computers.
Part 2: Buying advice.

    The Computers
History:
    Acorn was a very innovative British based company that appeared in the late 70s and finally folded in 1998. Among their legacy's are a long line of 8 and 32bit computers. The highly successful ARM processor core also started at Acorn.

Keeping things simple...
(There is MUCH more to Acorn than this)

8bit: Largely 6502 based.
    Acorn Atom - Great early 8bit. Very upgradeable. Many add-ons to be found. (Extended basic and GFX most likely) A rare and sort-after machine, so expensive.
    BBC - Commissioned by the British Broadcasting Corporation as a development of the Atom (via Proton prototype). At the time is was a very high-spec computer with a huge array of software and upgrades. Due to it's links with the BBC and great flexibility it was very popular in schools and many homes.
There were two versions: Model B and a cut-down version, the Model A (upgradeable to B spec). The B was pretty expensive in its day so the A was intended to help capture the home market. B's outsold A's by a huge margin so for the collector the model A is the one to get. The model B is a much better bet for a hobbyist as it will run far more software (games). Try to by a good-spec one with 5.25" FDD, memory expansion and extra roms. Most common among the many upgrades are: Internal HDD, speech, second processors, case upgrades, even compact flash! (Watford DFS is generally thought better than Acorn's - for compatibility you can have both!) The main criticism leveled at the 'Beeb' was the limited available ram so...
    BBC B+ - A development of the B. The main improvement was more memory (64-128k) A newer OS and improved ADFS were also included but can result in some (uncommon) software compatibility problems.
    BBC Master - Probably the most advanced 8bit personal computer ever sold! Largely compatible with the model B - both software and hardware. These are often ex-school models that have had a hard life but can come with an amazing array of extra goodies. A Master is a good choice if you want a BBC but have plans on it being a useful computer, so get one with as many extras as you can! (Don't laugh, Masters can still be found in some institutions running some big projects!)
The Master came in a few versions, most based on the same motherboard. Master 128 (basic) through to the Master 512 - DOS compatible, containing a '286 along with a 6502 variant. The Master Compact is an odd variant in a three box design with 3.5" FDD's. Interesting to the collector, but sufficiently different from the 'norm' to be of little practicable use to the hobbyist.
    The Tube - Is a BBC/Master upgrade port which allows for a multitude of options, mostly in 'cheese-wedge' shape along the lines of second processors : 6502+Ram, Z80+Ram, ARM+Ram, Teletext, to name but a few...
    Acorn Electron - A great little home micro. It's fundamentally a cut-down Beeb for the home market, oozing quality complete with the famous BBC basic. (the best version of basic ever made?) Sadly production difficulties made it late to market and it bombed. (Not helped by being perceived as somewhat incompatible with the model B) They are still not that sort-after. You can find 'fully-loaded' ones dirt cheep. Try to get one with a +1 and +3 interface for I/O and 3.5" floppy while you can! A ton of games and any other extras would be help too. A very, very underrated retro-computer.

    Don't forget: If you have an 8bit Acorn and don't have a copy of Elite there must be something wrong with you!
    Cool upgrade: Viglen case for the model B separating keyboard from main system box with FDD's built in. Room for HDD and other stuff too!

32bit:
    Acorn went through one of their may crises during this transition. They didn't really bother with 16bit as their 8bit ones were so good. In typical Acorn fashion they innovated to get out of the mire...
-New RISC processor - the ARM, at the time orders of magnitude faster than any other micro-processor and these days the ARM rules the embedded and PDA world.
-New OS - RiscOS was full of innovation and made the likes of MacOS and the awful Windows look neolithic from both a technical and functional point of view. Still in active development...
These computers are most easily split into three lines:
Pre-RiscOS 3.5
    The original OS (Arthur) on very early machines may be of interest to collectors, but for the retro-enthusiast who wants a functional computer its best to stick with RiscOS 2 and 3.1 machines.
    Archimedes A305, A310 - Base machines, probably with OS 2. Will hopefully run a good few games but not much else unless thoroughly upgraded. The A305 does not have enough memory as standard!
    Archimedes A400 series - Marketed as more serous computers. HDD's came as standard and they may well be found today heavily upgraded. Fundamentally similar to the 300's there were RiscIX versions which had a version of UNIX installed. The A540 was the top of this particular range and though superficially similar was a much more serious system from the point of potential specs - If you see one for sale, I'd suggest you buy, buy, buy! (especially if it's a well upgraded one - max ram 16mb)
    A5000 - For a long time Acorn's flag-ship computer. Faster, nicer, higher density FDD, HDD. Usually found with OS 3.1. I would suggest a well-specced A5000 with plenty of bundled software would be the ideal 32bit Acorn for the hobbyist. Well made, the widest compatibility with pre-3.5 software (especially games) and you may even find it useful! 4mb of ram (plenty) was top-spec for pre-3.5 computers but the A5000 can take 8mb.
    A4000 - A cut-down A5000 for home users. Nothing wrong with it and when they turn up they are usually dirt-cheap. Smaller case and not as upgradeable but more than capable of running all the available software of its era and a good alternative to the 'wedge-shaped' models...
    A3000, A3010, A3020 - Wedge-shaped computers in the traditional home computer sense. (think Amiga) Again a good choice for the retro-hobbyist, especially if space is tight. It's worth trying to get one with a HDD. The earlier A3000 is more upgradeable due to its larger case and motherboard design. With a HDD these make great retro games computers. All the best games of the day were very well ported to this platform and there were many amazing 'home-grown' ones too. They are also much easier to live with from a practicable point of view than, say, an Amiga A1200. You'll have much less trouble connecting one to modern monitors, and they are better made. (I own several Amigas by-the-way)
    A4 - Yes, Acorn did a portable computer. Basically an A5000 in an Olivetti laptop case, but pretty cool all the same - especially as it included another Acorn innovation: The best TFT dispaly tecknowlegy in the business - It makes even ordinary / cheap TFT's look really good! This IP went on to improve set-top-boxes (Acorn made a few of these too) and still crops up in modern systems. Sadly, though there were prototypes, Acorn never followed it up with a post-RiscOS 3.5 class machine.

Post-RiscOS 3.5
    OS 3.5 made its appearance with the RiscPC. I could eulogise endlessly about what an amazing machine this was/is but this is meant to be a quick guide...
Points to note: Compatibility problems do creep-in with older software and the RiscPC - primarily games. Such issues are not insurmountable, but it's best to view these as 'more serious' machines. They do have their own wide catalogue of software. (and games too!) For broadest appeal get a RiscPC600 or 700 with OS 3.5. It should run all but the most demanding new(er) software while managing most of the older stuff too.
These are multi-processor machines so many come with an x86 (33mhz 486sx - 133mhz AMD586). Depending on the x86 version and software DOS-Win98 will run making for a multi-purpose retro-box!
Very early RiscPC's had 8bit rather than 16bit sound but as they are quite collectible may be out of your price-bracket. Try to check for an included 16bit sound card with these. Interestingly the VGA output of early revision motherboards produces a significantly sharper display than later ones - It won't be an issue unless you've owned an early model. Acorn made modifications as a result of changing EMI regulations - it's fixable if you notice at all.
    Get one with lots of included software - much is still under development so you're in buy rather than the download territory of a 'dead' platform...
    Go for 2 rather than 1mb of V-Ram if you can - higher resolutions with more colours. A lack of V-Ram will cripple the system so stay away - unless you're looking at a good deal and can get some.
    A lot of RPC's have many, many extras. Watch a while to see where your price vs goodies ratio is.
Newer RPC's have much faster processors (StrongARM) and later OS (3.6 -> 4.02 / 4.39-Select v6). These are a good choice if you intend to 'use' these machines productively but backward compatibility will suffer.
    Some upgrades are highly sort after: Viewfinder GFX card. Unipod IO card. A RiscPC with these will be MUCH, MUCH more expensive than the rest.
    The case of the RiscPC is very cleaver. It can be upgraded with extra 'slices' if more internal room is required. New slices can be bought (at great expense) and the backplane only has room for two expansion cards per slice so keep that in mind when viewing a prospective purchase. It's a quite robust case but the spring-loaded flap(s) that cover the front drive-bay(s) easily break. (repairable)

    Acorn also made the A7000 - It will do pretty much everything a RiscPC will do, but slower. They can often be found cheap with a later OS version and a ton of software. Maybe not a good choice otherwise.

    This was the time other manufacturers began to make Clones - They do crop up. Your interest in these will lay very much in either the rarity or good-deal realm. RiscStation, Omega, to name but two...

Post-Acorn
    Just as Acorn were about to bring out the Phoebe (having spun-off all their best stuff as separate companies) they canceled it, renamed themselves element14, and then folded.
    In then end companies like Castle continued to make the RiscPC in faster form (the Kinetic) and updated clones. Castle have gone onto produce the Iyonix with RiscOS 5 - a thoroughly modern RiscOS computer.
RiscOS Ltd have a commercial business with a different (official?) branch of the OS (4.02-Select6)
Interesting mini-boxes like the A9home have also cropped up that make the Mac-mini look gargantuan!
    There's life in the platform yet!

Advice?
    If you're into things Retro, don't buy a post-RiscPC system - Being new, even on eBay, they go for a fair bit of money and will struggle running your retro-software.
If you're after a modern RiscOS system then by all means get an Iyonix from Castle (or eBay). Virtual RPC is also worth checking out along with the many emulators of earlier Acorn computers.
    You may even find an A9home will fit your home surprisingly well too!

    Oh, and the 'best' version of Elite ever made was for the 32bit Acorns.

    Important tips:
1) Most Acorn-made RiscOS computers have a 'non-standard' mouse. For a 'pre-3.5' machine a non-functioning mouse = a non-functioning computer - It's slightly less awkward with newer models. Replacement mice or adapters do turn up (rarely). The clones invariably use 'standard' keyboards and mice.
2) Most Acorn computers have a motherboard-mounted battery to keep time when switched off. Any such computer of a reasonable age may have a leaky battery - this is 'normal'. Trouble is that leaking battery acid will destroy the motherboard if left unattended - even small amounts!


    Buying advice
So you've seen the computer you want - what next..?
    It's important to understand the basics of eBay - people here can be split into two categories:

-Sellers - Want to get the best price they can.
    Most will be very honest but human nature dictates even the best of us may gloss over a few issues: Ask questions, BE POLITE!
    There are quite a few rogues out there. Check feedback closely. Watch out for auctions that say something like 'untested, no returns'. Often this is simply for self-protection (old hardware can give up at awkward moments) but this is a common way to dump known non-functioning kit on the unwary - especially Amiga kit!
    Where is the seller located? I'd honestly suggest NOT buying from anyone outside the EU, USA or Auz. (I'd also be very wary of auctions located in Italy). Why? The world is full of decent-honest people but you'd be amazed at the number of ways scammers have to separate you from your money.
     China and HongKong? You can get some amazing deals, but if it is too good to be true - it probably is.

-Buyers - Can be split into three groups:
1) Genuine buyers.
2) Snipers.
3) Bloody Idiots! (otherwise known to sellers as 'manor from heaven')
-sadly the vast majority - don't be one of these-
1) Genuine buyers.
    Someone who sees an item they want, makes a decision on what it's is worth to them, and puts in a maximum bid based on that. A reasonable plan: If this is you, fine. My advice to you is make your bid and DON'T return to the auction  until it's over!
Why?
-If you need to look you didn't enter the maximum amount you would be happy with in the first place and are in danger of joining group 3).
-Avoids the frustration of watching the price YOU will end up paying shoot up courtesy of the mindless-idiots in group 3).
2) Snipers.
    Someone who isn't bothered if they don't win and is interested in getting as good a deal as possible by making a bid at the last moment.
How?
-It reduces the time available to rival bidders for making counter-bids and cut's out the mindless-idiots in group 3) who won't have a 'target' to bid against.
-Rather than putting in your maximum bid, you put in a bid for what you regard as a good deal.
The down side is there may already be a higher bid, or another sniper leaves it later than you and steals the day.
3) Bloody Idiots!
    There's nothing good to say about such eBay'ers. (unless you are selling) It's a fairly common scenario:
You are a member of group 1). You've put in a high maximum bid to win, but obviously would like to buy for the lowest amount fortune will allow...
...then while you're watching the price starts to rise. One bid after another - all small increments but nothing to beat you. All the same, the item you were winning for £10.00 is now going to cost you £60.00! Arragh!
You look at the bidding history for the auction to find: Not lots of interest, but one lone person who has just made a series of 'hopeful bids'...
...The air turns blue with your cursing!
    Sniping reduces the likelihood of your falling victim to this scourge but if it's an item you badly want that's not always an option.
    It's no better if you are the bloody idiot either! It's likely you're not that bothered about winning if you're firing in such hopeful bids, so chances are someone has already put in a bid higher than you intend. Also after putting in 10+ unsuccessful bids you've given up BEFORE you reached your own personal maximum anyway!
In short: This strategy will get you nothing but curses!

I hope you've found this useful - it is a tremendously abridged version of the subject.

Useful Links? Here's a start:
http://www.old-computers.com/news/default.asp
http://www.mcmordie.co.uk/acornhistory/index.shtml
http://acorn.chriswhy.co.uk/index.html
http://www.iconbar.com/
http://www.acornarcade.com/
http://acorn.chriswhy.co.uk/Phoebe.html