Abraham Augustus Burton Jr. was born in New York City on March 17, 1971 and grew up in Greenwich Village, the first-generation American son of Belizean parents. Music arrived in the apartment before school did: his father was the superintendent of a building complex and spent Saturday mornings playing records loudly through a home-built speaker rig, his sister Carla took up the clarinet and eventually played Carnegie Hall, and his brother played drums. Burton wanted the cello and settled on the saxophone, a choice he has described as a matter of what a family could afford and of what a Black classical player could realistically expect in the early 1980s. He went to Fiorello H. LaGuardia High School of Music and the Performing Arts, where the head of jazz, Justin DiCioccio, put him in an all-star quintet that took best jazz combo from DownBeat in 1989 and won Burton a separate award as outstanding soloist. The same year he won the McDonald's Jazz Combo Competition in a trio with the drummer Eric McPherson, a classmate. Thirty-seven years later McPherson is still the drummer in his band. Two other LaGuardia classmates, the saxophonist Myron Walden and the bassist Dwayne Burno, came out of the same rooms.

School was only half of it. Burton spent his high-school years inside New York clubs and was taken up by the musicians he found there, among them Barry Harris, Billy Higgins, Elvin Jones, Max Roach, Richard Davis, Cecile Bridgewater, Bill Saxton and the drummer Freddie Waits. He has singled out one night watching Waits at Sweet Basil as the moment he decided to do this for a living. In the mid-1980s he met Jackie McLean and began an apprenticeship that turned into a formal one: he followed McLean to the Hartt School and studied there with him and with the drummer Michael Carvin, taking a bachelor's degree in music and performance from the University of Hartford in 1993, with honors. McLean remains the largest single presence in his playing and in his repertoire. Burton's first record opens with McLean's "Minor March," his second contains "Little Melonae," and he has never stopped programming his teacher's tunes.

His professional career began with Arthur Taylor. The drummer had reassembled his Wailers in New York after two decades in Europe, and Burton, still in his teens when the association started, held the alto chair for roughly five years. He is on both of the band's albums, Mr. A.T. (Enja, 1992) and Wailin' at the Vanguard, taped at the Village Vanguard on August 29 and 30, 1992 and issued by Verve the following year, and he toured Europe with the group and worked the Vanguard, Sweet Basil, Condon's, the Blue Note and Birdland with it. Taylor's death in February 1995 closed that chapter; the Jimmy Smith album Damn!, cut at the Power Station that January with Burton soloing on two tracks, carries a dedication to Taylor's memory on the sleeve.

Burton formed his own quartet in 1994, with Marc Cary on piano, Billy Johnson on bass and McPherson on drums, and Enja recorded it twice. Closest to the Sun was made at Systems Two in Brooklyn on March 11 and 12, 1994; The Magician was taped live at the Greenwich Village room Visiones on March 17 and 18, 1995, Burton's twenty-fourth birthday and the day after. The band toured Europe, Asia, Australia and the Virgin Islands and played the North Sea, Moers, JVC, Panasonic and Pori festivals. By the end of the decade the group had changed shape and name: as Forbidden Fruit, with James Hurt on piano and Yosuke Inoue on bass, it recorded Cause and Effect at Bleecker Street Studio on July 21, 1998, released by Enja in 1999 and billed to Burton and McPherson jointly. That record is where Burton moved from alto to tenor, the horn he has played almost exclusively since. The same quartet, with James Carter added as guest on three tracks, made Drifting Inward for the Japanese label M & I in the same year, this time billed to Burton and Inoue.

From the late 1990s he became one of the more heavily used tenor players in New York. He joined Louis Hayes's quintet for Quintessential Lou (1999) and has been in Hayes's bands ever since, across The Candy Man (2001), The Time Keeper (2009), Lou's Idea (2010), Return of the Jazz Communicators (2014), Serenade for Horace (2017, Blue Note), Crisis (2021), Exactly Right! (2023) and Artform Revisited (2024), a run of nine albums over a quarter of a century. He has been in the Charles Mingus organisation since 1998 and has played all three of its ensembles, the Mingus Big Band, the Mingus Dynasty and the Mingus Orchestra; he is one of the two tenor players on Mingus Big Band Live at Jazz Standard, recorded on New Year's Eve 2008 and winner of the 2011 Grammy for best large jazz ensemble album. Other sideman work from those years includes Horace Tapscott's Aiee! The Phantom (Arabesque, recorded June 1995), Dusko Goykovich's Bebop City (Enja, December 1994), Terumasa Hino's Off the Road (1997), Ali Jackson's festival record Groove@Jazz En Tête (2000), Santi Debriano's Artistic License (Savant, 2001, engineered by Rudy Van Gelder) and four SteepleChase albums for the trumpeter Jack Walrath between 2010 and 2015.

The most sustained partnership of his later career has been with the Romanian pianist Lucian Ban. Burton and McPherson are the front line and the engine of Ban's quartet Elevation, with John Hébert on bass, on Mystery (Sunnyside, 2013, from a January 2010 concert) and Songs from Afar (Sunnyside, 2016), the latter a setting of Transylvanian folk material with Mat Maneri's viola and the baritone Gavril Țărmure; Burton and Ban then reduced it to a duo for Black Salt, taped live at the Baroque Hall in Timișoara on June 5, 2018. He and McPherson have also kept their own name on records, with the self-released Future in 2011 and, as the Burton/McPherson Trio with Dezron Douglas, Live from Summit Rock in Seneca Village, recorded outdoors in Central Park on June 20, 2021 and issued by Giant Step Arts. He has been a fixture at Smalls in Greenwich Village since December 2007, and the club's archive lists 242 sets he has played there, 117 of them billed to his own quartet, quintet or trio. Away from the bandstand he took a master's degree in music at Rutgers in 2013 on a Ralph Bunche Fellowship, and now teaches there as a lecturer in jazz studies and director of the Rutgers Jazz Lab Band, alongside a chair at the Jackie McLean Institute of Jazz at the University of Hartford and a teaching post with Christian McBride's Jazz House Kids programme in Montclair. He is a Selmer artist and lives in Morristown, New Jersey.

Burton is a big-toned, high-pressure tenor player whose sound is loud before it is anything else, and the writing about him keeps reaching for images of heat and force: the New York Times said in 1996 that a cartoon of him would have flames coming out of the horn, and returned later to a tone it described as a strong but wavery yowl, capable of conveying a wounded tenderness in spite of all that volume. Grove's assessment is that the obvious influence is McLean, with elements of John Coltrane also evident, and that he is a highly capable ballad player who takes fast tempos with a controlled abandon. Both halves of that are audible. The McLean inheritance is in the slightly sharp, keening placement of the pitch and in the repertoire; the Coltrane inheritance is in the long modal ascents and in a preference for saxophone-and-drums duo passages that critics have compared directly to Interstellar Space, a comparison invited by the fact that Burton's steadiest partner has been a drummer rather than a pianist. The Belizean side of his background surfaces in the writing more than in the playing, in pieces like "Punta Lullaby" and "3rd Eye Punta" that borrow the Garifuna punta rhythm. He is a working composer as well as a soloist, and his own tunes, "Dad," "It's to You," "Corrida de Toros," "Deep Connection," carry across from his leader dates into other people's: Hayes recorded "It's to You" twice.