What features of a mountain bike's geometry gives it different characteristics?
The shorter the distance between the bottom bracket and saddle lowers pedalling efficiency, but increases standover height which gives you more room to throw the bike around.  This feature gets more pronounced as the riding style the bike is designed for becomes more aggressive i.e. an enduro frame will sacrifice some pedalling efficiency for stability over very rough downhill terrain.
A longer top tube leans the rider forward centring the weight distribution more, this gives better stability in the air and over rough terrain, but causes the bike to feel a little less manoeuvrable because it's harder to pull the front up.
Shorter chainstays allow the front of the bike to pull up easier because the pedals are closer to the rear axle (the pivot point when pulling the front up).  Aggressive bikes that want the stability of the longer top tube often shorten the chainstays to keep the bike feeling more nimble.
A higher bottom bracket lifts the centre of gravity up making the bike more difficult to corner, but a low bike is a bike that scrapes rocks more...
A steeper head angle results in a more responsive, lighter feeling steering.


So to give an XC example:


A pure XC bike will have a large BB to saddle height to maximise pedal efficiency, therefore a low standover height, but this isn't too much of a sacrifice as an XC bike is not aimed at very rough terrain.
Often rather than a long top tube, they have a slightly shorter one and use a longer stem.  This keeps the weight forward so the front doesn't keep lifting when pedalling hard up hill, but keeps the steering feeling light (not always desirable...).
Longer chainstays for the same reason as above.
Steep headangle.


Whilst an enduro:


Will have a low BB to saddle height so it can be thrown around over very rough terrain.
A long top tube for the extra stability.  An enduro bike can't use the long stem trick to minimise the top tube length, because long stems make the steering feel 'overactive' which is horrible on rough terrain.
Short chainstays to mitigate the long tube.
Slightly slacker head angle because it makes the bike more stable when going downhill.


Most MTB frame styles have roughly the same BB height, as everyone wants a better cornering bike and it's difficult to pedal over rough terrain anyway.  The only exception I think of are pure downhilling frames, where a very low BB height is desired but to minimise pedal smashes, shorter crank lengths are the norm.

Headangles tend to quite steep on most MTB frames even on very different styles, for example XC and dirt jump frames have similar headtube angles.  Once again the notable exception are downhilling frames, which have very slack headangles; although this for improving stability when on steep descents (it's harder to go over the bars).

The above is an oversimplification.  In the real world the parts play off each other, and their characteristics will change as a bike moves through the rear travel (if it's got any of course).  Also the above are frame characteristics; the length of stem, width of the bars, length of cranks, height of forks, etc. all play their part in the overall feel just as much as the frame does.