Bumpmap textures are textures that can add a sort of 3D, textured effect to textures.
The icon texture, as far as I know as I have never owned one, is a texture that lets you upload a small image. It appears under your name ingame.
Header, logo, splatter and background textures are textures that allow you to change how Toribash looks when it starts up. Normally you have the Toribash logo, but the logo texture allows you to change it to whatever you want.