A lot of jazz fans get scared off John Zorn because there's so much of it and in such widely divergent styles, but if we're talking about the great jazz albums of the 21st century then I think some space should be made for:
Masada, 50th Birthday Celebration: Volume 7 (recorded September 18th 2003 at Tonic) - This is Zorn's quartet of alto sax, trumpet, drums and bass playing themes on Jewish scales. It's probably my favourite jazz album not to feature piano.
John Zorn, Nova Express (recorded October 2010 in the studio) - A quartet of piano, vibes, bass and drums. Although this shows more of the influence of Zorn's "game" style (slightly more quirky and idiosyncratic but nowhere near "Emperor's New Clothes" territory), it's for the main part a very solid acoustic jazz set and was my Album of the Year last year.
Stepping away from Zorn now:
Brad Mehldau, Live in Marciac (recorded August 2nd 2006 at the Marciac Jazz Festival). There are lots of great recordings by Mehldau, and the two solo piano live albums (this one, and 2003's Live in Tokyo) both deserve a place in anyone's jazz collection. This was my runner-up for Album of the Year last year.
Gwilym Simcock, Blues Vignette(2009) Simcock is a young British pianist who performs with Acoustic Triangle (a band well worth hearing but too dry for my taste) and has garnered huge critical attention for his two (more or less) solo albums. For me, this one has the edge over Good Days At Schloss Elmau, which was a nominee for the Mercury prize last year. Very much in the ECM/Keith Jarrett tradition.
One recent album I really enjoy is Pat Metheny's Orchestrion:
A unique album of ensemble-oriented music using acoustic instruments that are mechanically controlled using solenoids and pneumatics. Takes the "solo record" concept to a whole new level. Watch the videos here to get a sense for what's involved:
Some artists I like that I think haven't been mentioned yet:
Ken Vandermark, especially his School Days and Atomic/School Days projects although the Vandermark 5 and DKV bands are also worth hearing
Atomic, young Scandinavians who sound nothing like the ECM Scandinavian musicians
Daniel Levin, outside jazz cello groups
Kris Davis, piano
Tony Malaby, tenor
Harris Eisenstadt, drummer
Peter Evans, trumpet
Henry Threadgill, reeds born in 1944 and still making great music
Michael Moore, alto sax and clarinet An American expat who has been living in the Netherlands since the 1980s
Alexander von Schlippenbach, piano born in 1938 and still making great music, check out Monk's Casino and his long running trio with Evan Parker
The Portuguese Clean Feed and Spanish Fresh Sounds New Talent labels are to 21st Century jazz what HatArt, ECM and Black Saint were to the last two decades of the 20th Century.
I used to find out about all kinds of great newer musicians by listening to an online station called Barcelona Jazz Radio but, unfortunately, they seem to have gone kaput. http://www.barcelonajazzradio.com/
A lot of the best, most listenable jazz comes out of Europe. I've always liked the ECM label. There are still American stalwarts like Keith Jarrett and some younger guys like Chris Cheek and Stefon Harris but, for the most part, the Europeans sound better to my ear these days. They're not afraid to sound "pretty" and melodic while still being intelligent. It seems like many American jazz musicians, by contrast, either pursue relentlessly sour, atonal, doodlebugging and noise to show how "pro" they are or they wind up going to the opposite extreme and go the Kenny G route.
I just discovered a band called Seatbelts. They're mainly Jazz/blues but have a ton of stuff in a lot of genres. Not sure if they've already been mentioned or not but I downloaded their FLAC discography and I absolutely love the diversity of everything so far.
Also it's really not Jazz music but there's an electronic/edm artist that has these crazy saxophone solos in the middle of most songs, Big Gigantic if you're interested.
I just recently joined this forum and just discovered this thread. There are some great suggestions that I look forward to trying out. For me, some of my favorite 21st century albums are "The Gnostic Preludes" and "O'O" by John Zorn, "Infernal Machines" by Darcy James Argue and both Floratone albums.