What recent discoveries have amateur mathematicians made?

About ten years ago Ahcène Lamari and Nicholas Buchdahl independently proved that all compact complex surfaces with even first Betti number are Kahler. This was known since 1983, but earlier proofs made use of the classification of surfaces to reduce to hard case-by-case verification.

At the time, Lamari was a teacher at a high school in Paris. Apparently he announced his result by crashing a conference in Paris and going up to Siu (who had proved the last case in the earlier proof in 1983) with a copy of his proof. Lamari's proof was published in the Annales de l'Institut Fourier in 1999 (Courants kählériens et surfaces compactes, Annales de l'institut Fourier, 49 no. 1 (1999), p. 263-285, doi:10.5802/aif.1673), next to Buchdahl's (On compact Kähler surfaces, Annales de l'institut Fourier, 49 no. 1 (1999), p. 287-302, doi: 10.5802/aif.1674)

After Martin Gardner published what one mathematician claimed to be a complete list of convex pentagons that could tile the plane, amateurs (Richard James III, a computer scientist, and Marjorie Rice, who had no mathematical training beyond high school) discovered several more classes of pentagons that could tile.

Greg Egan. He's a very renowned science fiction writer who holds a bachelor degree in mathematics. He wrote, as a coauthor, 2 articles which were published in peer-reviewed journals, one of them is with John Baez. The first one was written when he was approximately 40 years old.

There's also more eccentric example of Andrew Beal, which is much more known in the world of poker. He made however one minor conjecture in number theory for whose proof or disproof he offers $100,000.

And there's also a list on wikipedia which might be worth going through.

Edit: (nov-2018) Some recent progress by G. Egan has been made with an anonymous 4chan-member, on a problem on permutations.