Glossy finish is the shiny, rich-looking paper most people are traditionally used to. It tends to show fine detail better, but also shows fingerprints better.
Matte finish is a highly textured print finish that resists fingerprints and scratches. It is much more dull, and does not reflect light, Many photographers use matte finishes for wallet prints because they usually get handled more than normal. Some also use it for poster-sized prints to reduce glare.
Lustre finish is not as shiny as glossy, but not as flat (dull) as matte finish paper. Lustre paper has a textured finish (some call it "pebbled" that hides fingerprints. Most portraits and wedding photos are typically printed on Lustre finish paper.
Metallic finish is a high gloss finish with very saturated colors. It appears as though the image is printed on metal (It almost compares to the back side of aluminum foil). Images tend to really "pop" when they are done on metallic paper. However, note that "highs" (bright areas) have a tendency to be "blown out" and over-highlighted on metallic paper.