I've heard that the two main methods of 3D character animation are skeletal animation and vertex mesh animation. Could someone explain to me what they think the strengths and weaknesses are of each of these? I'll be using blender to create the models if that helps.

With mesh keyframes you have to handle alot more data (entire meshs are replicated in the new positions for each frame). Data has to be piped into the GPU usually with tweening for the rendered frames between the keyframes.

With skeletal the data is the basic mesh in a neutral position and all the vertice positions have to be calculated (matrix transformations) on the fly from canned angle rotation(and some translations). You can also do free form like ragdoll physics on the fly (creating new angle data on the fly). The data flow into the GPU can be less if the transformations are done there