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Tyres
Width vs Length
It is generally regarded that the wider the contact patch is, the greater its limit lat-acc - the better it is at cornering. The longer it is, the greater longitudinal performance it is likely to have. As with anything to do with tyre, this is not a cast iron rule, but is generally true.
The contact patch area is a deeper subject. Throw away statements like "more rubber on the road = more grip" are basically flawed (although in some ways also accurate). Confused yet?
Contact Patch Area
The amount of "grip" an inch of rubber creates when pressed to the tarmac increases the harder you press it onto that piece of road. Ignoring chaotic complications, what this means is that, theoretically, if you had 50Nm of force to distribute across the whole contact patch, it wouldn't matter if the contact patch area was 10 square cm, or 100 square cm, the total "grip" would be the same, its just that each square cm produces more grip in the 10 square cm example because each square cm receives more of the load.
Now it isn't quite that simple, but again, it isn't far off. The big difference, however, is that a tyre with a greater contact patch area shares the load out across the tyre tread more evenly which means less heat in the tyre and less wear.
Affect of PSI on the contact patch
A common myth about tyres is "wider tyres mean you get more rubber on the road". This again, isn't necessarily true. The area of the contact patch is a function of the load on the tyre, and the tyre pressure alone. It is true that other factors can come into it, like the sidewall stiffness, etc but in normal operating conditions it is true that pretty much nothing other than pressure and load affect the area of the contact patch to any significant degree.
 What a wider tyre will generally mean is that it had a shorter, wider contact patch. Narrow tyres will have a narrower, longer contact patch. But if you put both tyres on the same car and set both tyres to the same pressures, the actual contact patch area would be approximately the same.
on to the affect of vertical load (why can lighter cars corner faster?) ...
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