Artist's proofs are a special subset of the regular limited edition. They usually sell for 10-30% more than the regular edition. Artist's proofs began back when limited editions were all hand-pulled from a one-man litho press. The artist's proofs were the first prints pulled off a fresh "stone" (the plate which was drawn or etched by the artist to create the prints). The stone wore down as the number of prints was increased, so that the APs were the sharpest and most colorful of the lot. Thus, they sold for more, being a better product.
These days, in the world of modern offset lithography, all the prints in an edition are nearly identical. Artist's proofs maintain their collectability and value because A) They are a small subset of the edition, and B) they usually come directly from the artist. In these days of mass marketing, getting something directly or even indirectly from the artist is rare. Publishers usually turn over the artists proofs to the artist for his or her own use, and retain all the other prints in the edition.
How rare? it varies. Traditionally, artist's proofs number 10% or less of the total edition, so if there is an edition of 500, usually there will be only 50 artist's proofs. In other countries, it may be more or less. In Russia, for example, the artists' proofs may run 30% of the edition.
Artist's proofs are signed "A/P" or "Artist's Proof" and may or may not be numbered with the serial number of that portion of the edition.
Artist's proofs are generally considered a status symbol in the world of art collecting. And their resale value is proportionally higher than the numbered editions.