Quote: "What the heck is "vertex weight" and how does it affect the animation? "
Basically, in bone based animation, each vertex is assigned a bone which will control the vertex as the bone moves. the 'weight' value assigned to that vertex for that bone is how strongly the bone's position, rotation, and scale will affect the position, rotation, and scale of the vertex.
each vertex can be assigned multiple bones, but the total weight for all bones on any given vertex should normalize to 1.0
a simple example: lets say you have an arm. it has 2 bones: upper arm and lower arm. vertices in the middle of the upper arm will be assigned fully and solely to the upper arm bone, at a weight of 1.0
when the lower arm moves, the lower arm bone has no effect on these vertices. but now you get to the elbow, it is part way between the two bones, in order to get a smooth deformation, you will want to split the vertex weights between the bones say upper arm 0.5, lower arm 0.5 for vertices in the exact middle edge loop, then say 0.75 and 0.25 for the next loop out in either direction.
most animation software has tools for automatically weighting vertices based on proximity to bones, or by user defined groups and the like, though you will usually need to go in and do some touch up work on individual vertices to get the best results.
