Nicholas Aaron Finzer was born in Rochester, New York, on May 22, 1988, and grew up in the town of Hilton just northwest of the city. Neither parent was a jazz musician, though his mother, the flutist Sherry Finzer, had played through high school and picked the instrument back up once her son started playing. He sang and picked at the family piano as a small child, and when the band teacher came round at the start of fourth grade he asked for drums, with trumpet as his second choice. He was tall for his age and was assigned the trombone instead. What converted him was a sound he heard a few years later: the Eastman School of Music's trombone choir, twenty-five or thirty players at once, rehearsing a mile or two from where he lived. He began formal lessons in sixth grade with a local band director and worked his way up through Rochester teachers to Mark Kellogg, one of Eastman's trombone faculty.

The decisive turn came in tenth grade, when he joined a year-round jazz orchestra run through Eastman's community division that was built around the music of Duke Ellington for Jazz at Lincoln Center's Essentially Ellington festival. He had intended to be a classical trombonist; Ellington ended that. Still in high school he met Wycliffe Gordon, then in the Lincoln Center Jazz Orchestra, when Gordon's band passed through Rochester, and talked his way into lessons in New York. Gordon went on to write four pieces for him and has recorded some of them. Finzer took his bachelor's degree in Jazz and Contemporary Media at Eastman, then a master's at Juilliard, graduating in 2012, where he studied with Steve Turre. He was a finalist in the International Trombone Association's Carl Fontana Jazz Trombone Competition in 2010 and won the Eastern Trombone Workshop's National Jazz Trombone Competition in 2011.

His recording debut arrived on January 22, 2013 with Exposition, a sextet date of his own compositions that was also the first release on Outside in Music, the label he founded to put it out. Two of its pieces won him ASCAP Herb Alpert Young Jazz Composer Awards, in 2013 and again in 2015. The band settled quickly into a fixed personnel that has now lasted more than a decade: Lucas Pino on tenor saxophone and bass clarinet, Alex Wintz on guitar, Glenn Zaleski on piano, Dave Baron on bass and Jimmy Macbride on drums. The Chase, an album-length work written in 2014, came out on Origin Records in July 2015, earned a feature in DownBeat and reached number six on the JazzWeek chart. The sextet took its name from his third record, Hear & Now, released February 17, 2017.

Alongside the leader dates he built a working New York sideman life. He has held the trombone chair in Lucas Pino's No Net Nonet since its first recording in August 2013, and the band's monthly residency at Smalls in Greenwich Village runs in the club's own performance archive from November 2012 through 2019. He worked in tubist Bob Stewart's group, in Ryan Truesdell's Gil Evans Project, in Eyal Vilner's big band, and from 2016 in the Anat Cohen Tentet, whose two Anzic albums, Happy Song and Triple Helix, were recorded at Avatar Studios in August 2016 and at Sear Sound in January 2019. Triple Helix was nominated for the Grammy for Best Large Jazz Ensemble Album. He has also played with Wynton Marsalis's Jazz at Lincoln Center Orchestra, with Dafnis Prieto, with Frank Wess, Terrell Stafford, Frank Kimbrough, Carl Allen and Walt Weiskopf, and toured North America, Australia, New Zealand and Europe with Scott Bradlee's Postmodern Jukebox. His touring has taken him to Carnegie Hall, Dizzy's Club, the Sydney Opera House, the Bimhuis, Birdland, the Blue Note and the SFJazz Center.

In 2018 the University of North Texas created a jazz trombone professorship and gave it to him, the first the school had had; he is now Associate Professor of Jazz Trombone there, and was in residence at the Hochschule für Musik Würzburg in 2025. He had previously been a visiting professor of jazz trombone at Florida State University, a sabbatical replacement at Hunter College, and a teacher at Jazz at Lincoln Center's Jazz Academy. Outside in Music has meanwhile grown well past its origins as a vehicle for his own records: it won a Grammy for Best Vocal Jazz Album in 2024 and drew two more nominations the following year, both for the Dan Pugach Big Band. His own catalogue has kept widening. Cast of Characters (2020) is a suite of character portraits; Out of Focus (2021) is a lockdown record of overdubbed trombone choirs and remote quartets with no Finzer compositions on it at all; Dreams, Visions, Illusions (2023) came out of a 2020 Chamber Music America New Jazz Works grant; Legacy (2024) marked the centenary of J.J. Johnson with three of Johnson's own former sidemen and topped the JazzWeek chart for three consecutive weeks that June; and The Jazz Orchestra Volume 1 (2025) re-cast his sextet book for a seventeen-piece band. DownBeat's critics named him Rising Star Trombonist in the 2020 poll.

Finzer names three trombonists as his foundation, J.J. Johnson, Curtis Fuller and Slide Hampton, and two mentors, Steve Turre and Wycliffe Gordon, who has called him "a new voice [in] the pantheon of upcoming trombone greats in the making." The tone is bright, tightly centred and articulated hard, closer to Johnson's clean bebop line than to the plunger tradition, though he goes looking for that tradition deliberately: his reading of Ellington's "The Star Crossed Lovers" on Out of Focus, scored for five overdubbed trombones, reaches for the vocalised plunger sound of Joe "Tricky Sam" Nanton. What separates him from most trombone leaders is that the writing came first and has stayed first. He is a suite composer, and the shape recurs: The Chase was conceived as a single album-length work, Cast of Characters as a gallery of portraits, Dreams, Visions, Illusions as three linked movements. The model is Duke Ellington, and specifically Ellington's habit of writing for the individual rather than for the chair, which is why the same six players have appeared on nearly every record he has made. Beyond the trombone he cites Miles Davis, John Coltrane, Herbie Hancock, Chick Corea, Pat Metheny, Joshua Redman and Michael Brecker as players who kept pushing their own envelope, and among large-ensemble writers Gil Evans, Bob Brookmeyer and Maria Schneider.