I'm 100% sure all of the following Wii games have bump-mapping/normal mapping:

1st party:
Super Mario Galaxy
Metroid Prime 3: Corruption
Super Smash Bros. Brawl
Wii Sports Resort
Battalion Wars II

3rd party:
The Conduit
Resident Evil 4: Wii Edition
Resident Evil: The Umbrella Chronicles
Dewy's Adventure
Deadly Creatures (not too sure)

The shiney parts of The Conduit are due to what's called gloss mapping. It makes textures that should reflect light (just imagine an object coated in gloss paint) reflect light. Diffuse mapping, which The Conduit also uses, is pretty much the opposite. A diffuse mapped texture will reflect light in the same way that an object coated in matt paint would. I'm pretty sure I've got that right but I will stand to be corrected.

Bump mapping makes an item appear more realistically in 3D, so that light reflects off variations of the surface realistically, and combined with diffuse and gloss mapping makes models such as the enemies and weapons in The Conduit appear to be in 3 dimensions rather than 2. Hope that makes sense, I'm not vey good at explaining things.

The best way to understand bump maps is to actually get your hands dirty. There's a 30 day trial of a program called Crazy Bump that you can download from http://www.crazybump.com/here. It allows you to apply bump maps to textures (or any image file) and see what the effect is.