Fine art paper is usually made using 100% cotton rag content, is most often acid-free and therefore is suitable for archival purposes since it addresses the problem of preserving documents for long periods (see "How Long Will Your Photo Prints Last?" at PhotoShelter Blog for example). It qualifies as matte paper, thought it certainly has a texture, a look and tactile experience that is simply lacking in photo papers, which are constructed on a plastic base. It can be an acquired taste, I would recommend you buy a sample kit or a small amount of paper before, just to try. When should it be used? It's an option you may want to consider if you are selling your prints, for example. Galleries and museums generally favor prints on archival paper too.

Luminous Lanscape had a review back in 2007 of high-quality Baryta-based fine art paper ("These papers are akin to the papers used for various non-photographic fine art processes"). It mentions the Hahnemuhle Fine Art Baryta 325, Harman Gloss FB AI, and Ilford Galerie Gold Fibre Silk. You certainly can't go wrong with anything coming from a very old, respectable company like Hahnemuhle. Less expensive, I would also recommend Red River's Aurora Fine Art paper, I actually made two prints on both the Natural and White variations of that paper yesterday, and they look pretty good (I would recommend White). I'm planning on buying some Ilford Galerie Gold Fibre Silk, I'll update this answer accordingly.

A few more things to keep in mind if you end up using high quality Fine Art Paper:

• make sure you download the ICC profile from the paper vendor to optimize your print. Here are Red River's, and Ilford's. Said profile is a fine-tuned combo associating your printer to their paper. There is often a PDF in that bundle that will tell you which settings to use for your printer, specifically for that paper.
• many semi-pro printers (I've an Epson R2880) require you to use a different black ink for matte vs. glossy/satin paper, don't forget to swap cartridges if needed.
• If you are using matte or fine art papers with pigment inks, an interleaving sheet is highly recommended for print storage or transport. Black pigment inks rub off and scuff easily on matte and cotton papers.
• if you print on archival paper, make sure your mats and backs are also archivals, i.e. 100% cotton. I would recommend Matcutter.com for example, here is a high quality archival cotton, buffered, 4 Ply, mat, though you can customize to your needs.
• if you are going to store prints in plastic bags or sleeves, look for plastic material that is made from biaxially oriented polypropylene. Avoid using any materials that contain polyethylene or plasticizers which can cause rapid yellowing of your paper.