Is there a specific kind of caulking I have to use to cover up the little gap between my vanity marble backsplash and the wall?

A "bathroom" caulk will work fine. (should say like Bathroom, tub/tile, etc.. )

Basically any type that can get wet.

Just make sure it is not adhesive caulk, a silicon caulk is best as it is mildew resistant.

Dap makes Qwik Seal plus which contains mildewcide. DAP Products - Caulks, Sealants, Adhesives, Repair and Floor Covering It should be good for that crack as well as the gluing on the back.