Glossy finish is very common today at fast, local photo finishers as well as online printing retailers. These photographs have a shiny appearance which does quite well showing the details and colors of the image, but can create a glare when reflected off light, and are highly susceptible to fingerprints.
Matte finish is what you have most likely viewed if you have ever noticed a "texture" on a photograph. While this texture does not detract from the detail and the color of the image, one of its purposes is to help reduce glare by deflecting light. Additionally, this texture helps reduce the fingerprint problem normally associated with glossy prints. Matte finishes are a good option for photographs which are frequently handled and/or viewed outside of a frame or album.
Luster finish is most easily thought of as being somewhere in between gloss and matte. Luster prints are slightly glossy, which helps enhance the image's color and details, but also feature a very fine "texture" to help reduce problems associated with glare and fingerprints. A Luster finish is a popular choice of professional photographers, particularly with wedding and family portraits, as it offers a nice balance between the features of glossy and matte.