3d art, long explanation
Actually, let me be a nerd and explain what weight painting is. OK, you have a sweet mesh, and a skeleton of bones for that mesh. You need to specify how much the bones affect the mesh when they bend, that is their weights. If we're lucky, you can use automatic weights. I wasn't lucky.
So I'm literally painting these weights on the mesh, which can make for nicer animations since doing it by hand lets me have more control.
3d art, short explanation
In short terms, the mesh is blue, and I'm making each little part red in such a way that when the bone bends, the mesh bends too.