in general if you hear about weights it usually means that something will be associated with a number value, usually between 0 and 1.
So if you have vertex A and vertex B, and you have Arm and Foot, if vertA has 0.2 weight to Arm and 0.8 weight to foot, it will go more towards the foot. And vertB has weights like Arm = 1 and foot =0, therefore it goes all the way to the arm.
With rigging and skinning you assign vertices to joints like this. And in texturing you'll sometimes use a black and white texture map to define weights. 0 = black and 1 = white, and 0.5 = gray.
And vertices can also be assigned color values and this is often used in environment art as a way assign materials in a layered way.
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in max it can be used for affecting subdivision.....
I think there are other modifiers that use it but I can't remember which ones at the moment
added programmatically to seams of procedural object so to affect a mesh smoothing modifier