What do the number mean? Firstly understand your product!
Binoculars have a two number code that defines the basics i.e. 8x40.  These numbers mean that a binocular will appear to bring a viewed object 8 times nearer and they have an object lens (the lens nearest the object you are viewing) of 40mm diameter. 
What is the best object lens size to go for?
The size of the objective lens is important because it is an indicator of how much light can enter the binoculars and therefore your how much light enters your eyes. The more light the better the binoculars will perform at dusk and dawn.  However, in good bright daylight most casual users will not be bothered by the lower performance of a smaller objective lens.  The generally offered size of object lens is anywhere between 20mm and 60mm.  The smaller size normally means the binocular will be a pair of pocket sized  binoculars for occassional use and the larger size can make the binoculars heavy and unmanageable unless used on a tripod. 
Why would I buy binoculars with small objective lenses?
Top quality glass coatings and top quality prisms can mean that top quality manufacturers can out perform cheaper models even with smaller lenses. As previously stated, unless you are a serious user, such as a birdwatcher, the low light performance will be of less importance and the lightness and convenience of a physically small instrument will be top of your list.
What is the best magnification?
There is no such thing as a best magnification.   Many people are seduced by high magnification...... don't be!  The problem with high magnification is that the image will shake no matter how steady you are.  It is generally thought that the highest hand held magnification should be 10x, i.e. objects appear ten times closer.  At the lower end, magnifications of less than 6x are generally reserved for such things as opera glasses etc.  The recommended magnifications for nature, sport, holidays and general viewing etc should be from 7x to 10x.  If star gazing or sea watching is your thing then higher magnifications can be used but you must use a fixed mount such as a tripod. 
What should I avoid?
Personally I would never buy zoom binoculars, there is no point zooming from 7x to 10x and anything that magnifies more is useless.  Don't buy ruby coated lenses.  These are only ever applied to cheap optics as are yellow coatings.  I have not come across other colours but I can guaranteee that ruby coatings make things appear red and yellow ones make things yellow. Don't be taken in by "the most powerfull binoculars you can buy" because it is a load of flannel.
Finally, what should I buy?
Go for something between 7x20 and 10x25 for pocket binoculars and for general use go for someting between 7x35 and 10x50.
Things to look for that are always present in good binoculars are: Multi coated optics, centre focussing, individual eye focussing (so that the optics can be adjusted to the differences between your two eyes).  Options that are beneficial are : gas filled  or nitrogen sealed (so that they don't fog), fold down eye cups (for spectacle wearers, though some older top quality instruments do not have this), rubber armouring will protect from knocks and small dinks (though nothing will protect instruments dropped from a great height.)

Roof prism binoculars (where the binoculars are in line like two tubes) perform less well than the equivalent porro-prism binoculars (where the lenses are offset like older traditional patterns.)  However, modern preferences means that the roof prism patterns  are more popular nowadays and those roof prisms made by top quality suppliers can out perform cheaper porro-prisms. You pays your money.............

A list (just my opinion) of some of the manufacturers who I would recommend  are LEITZ, ZEISS WEST, ZEISS JENA (EastGerman)  OPTOLYTH,  OPTICRON, SWAROWSKI, SWIFT, NIKON, PENTAX, BAUSCH & LOMB.  There are quite a few more, apologies if you think I have missed some out.... sorry, I probably have.  However, as a novice, if you buy from one these manufacturers you will not go too far wrong.  I would always recommend that you try the instrument first.  I tried the best instruments from all of the above manuacturers before settling on my Leitz and it comes down to personal preference when chosing between the cream. The better models are truly incredible.

REMEMBER, ALTHOUGH TRUE FOR ALL THINGS IT IS ESPECIALLY TRUE FOR OPTICS.  IF IT LOOKS TOO GOOD TO BE TRUE.... IT IS! AND YOU WILL ONLY GET WHAT YOU PAY FOR!