Glossy Photo Paper
Glossy photo paper is a bright white paper that is coated to give it a shiny and glossy finish. Colors are more vibrant and vivid. Additionally, if your images were in focus when you took the pictures, the photos print crisp images. There are multiple finishes available, such as high gloss, gloss, soft gloss or semi-gloss, which indicate the degree of shininess of the paper.
Matte Photo Paper
Matte photo paper isn't coated with gloss, so glares and reflections aren't as noticeable. You are less likely to detect smudges and fingerprints on matte paper. The images appear a bit duller than with glossy paper, because matte paper isn't as shiny, but images on matte paper can be as clear for certain photos. Additionally, because the matte paper is duller, it is better for printing text than photos. More advanced products have improved the image quality of photo prints.
Tips
When printing with either paper, wait for the photo to dry completely. That decreases the chance of getting smudges and fingerprints on your photos.