In the early days of the band, Wright dabbled with brass before settling on Farfisa electric organs as his main instrument onstage. He originally owned a single-manual Combo Compact model, which was used for early recordings of "Interstellar Overdrive", and later upgraded to a dual-manual Compact Duo. During the 1960s, Wright relied heavily on his Farfisa fed through a Binson Echorec platter echo, as heard on the Ummagumma live album. On later tours, the instrument was fed through a joystick control allowing the signal to be sent through up to six speakers in an auditorium, which was called the "Azimuth Coordinator". Wright stopped using the Farfisa after The Dark Side of the Moon, but revisited it in later years, playing it on Gilmour's On An Island tour. It was recorded for the sessions that eventually became The Endless River.

Wright, whose father was head biochemist at Unigate Dairies, grew up in Hatch End, Middlesex, and was educated at the Haberdashers' Aske's School. He taught himself to play guitar, trumpet and piano at age 12 after he was recuperating from breaking a leg. His mother helped and encouraged him to play the piano. He took private lessons in musical theory and composition at the Eric Gilder School of Music and became influenced by the trad jazz revival, learning the trombone and saxophone as well as the piano. Uncertain about his Future, he enrolled in 1962 at the Regent Street Polytechnic which was later incorporated into the University of Westminster. There he met fellow Musicians Roger Waters and Nick Mason, and all three joined a band formed by classmate Clive Metcalf called Sigma 6.

Wright had been fond of the Greek islands since a sabbatical visit in 1964, before Pink Floyd were formed. He moved to Greece in 1984 after Zee, briefly retiring from music, and enjoyed sailing and yachting. In his later years, Wright lived in Le Rouret, France, and spent time on a yacht he owned in the Virgin Islands. He found sailing therapeutic, relieving him from the pressures of the music Business. He was also a keen collector of Persian rugs.

Pink Floyd had stabilised with a line-up of Barrett, Waters, Mason and Wright by mid-1965, and after frequent gigging that year became regulars on the underground live circuit in London. While Barrett was the dominant member, writing most material, singing most lead vocals and playing lead guitar, Wright played a supportive role, playing keyboards and singing, with occasional lead, and writing his own material. As the most qualified musician, Wright was responsible for tuning guitars, and would often tune Waters' bass for him in concert. Later on, he had a Strobotuner to tune guitars silently during gigs. In the band's early days, before acquiring a full-time road crew, Wright was responsible for unloading the gear at the end of each gig.

Wright grew up in Hatch End, Middlesex and met Future Pink Floyd bandmates Roger Waters and Nick Mason while studying at the Regent Street Polytechnic. After being joined by frontman and Songwriter Syd Barrett, the group found commercial success in 1967 before Barrett's instability led to him being replaced by David Gilmour and Wright taking over songwriting duties with Waters. Initially a straightforward singer/songwriter, Wright later acted as an arranger to Waters and Gilmour's compositions. He began to contribute less towards the end of the 1970s and left the band after touring The Wall in 1981. He rejoined the band as a session player in 1987 for A Momentary Lapse of Reason, and became a full-time member again for The Division Bell in 1994. Sessions with Wright during this period were later released on the album The Endless River. Away from Pink Floyd, Wright recorded two solo albums, including a collaboration with Anthony Moore on Broken China, and briefly formed the duo Zee. After rejoining Waters, Mason and Gilmour as Pink Floyd for Live 8 in 2005, he became part of Gilmour's regular solo touring band, singing occasional lead on songs such as "Arnold Layne", before his death in September 2008.

While not credited for vocals on The Piper at the Gates of Dawn, he sang lead on Barrett-penned songs like "Astronomy Domine" and "Matilda Mother". Examples of his early compositions include "Remember a Day", "See-Saw", "Paint Box" and "It Would Be So Nice". Wright was close friends with Barrett, and at one point the pair shared a flat in Richmond. After Barrett left the group in 1968 owing to mental health issues, Wright considered leaving and forming a group with him, but realised it would not have been practical.

Wright played the piano and Hammond organ in the studio from the start of Pink Floyd's recording career; using the Hammond's bass pedals for the closing section of "A Saucerful of Secrets". He used a Mellotron in the studio for some tracks, including Ummagumma's "Sysyphus" and on the "Atom Heart Mother" suite. For a brief period in 1969, Wright played vibraphone on several of the band's songs and in some live shows, and reintroduced the trombone on "Biding My Time". He started using a Hammond organ regularly on stage alongside the Farfisa around 1970 and a grand piano became part of his usual live concert setup when "Echoes" was added to Pink Floyd's regular set list. All three keyboards are used in the concert film Pink Floyd: Live at Pompeii.

In the 1970s, Wright began using synthesizers such as the VCS 3, ARP String Ensemble and Minimoog, which were featured on "Shine on You Crazy Diamond". Wright wrote the closing part of the track alone, and included a brief extract of the band's early single "See Emily Play" on the Minimoog towards the end. He used a number of electric pianos during the 1970s, including a Wurlitzer fed through a wah-wah pedal on "Money" and an unaccompanied Rhodes introduction for "Sheep" on Animals.

Following Barrett's departure and replacement by David Gilmour, Wright took over writing duties with Waters but gradually became less involved as a singer and Songwriter as the band's career progressed. His organ playing remained an important part of the band's live set, including "Interstellar Overdrive", "Set the Controls for the Heart of the Sun" and "Careful with That Axe, Eugene" and he contributed musical themes for film scores (More, Zabriskie Point and Obscured by Clouds). He made significant contributions to Pink Floyd's long, epic compositions such as "Atom Heart Mother", "Echoes" (on which he harmonised with Gilmour for the lead vocals) and "Shine On You Crazy Diamond". On 1973's The Dark Side of the Moon he composed the music for "The Great Gig in the Sky" and "Us and Them". He also contributed to other album tracks such as "Breathe" and "Time", singing the lead vocals on the latter's chorus.

Wright recorded his first solo project, Wet Dream, in early 1978 in Super Bear Studios, France, which featured touring Guitarist Snowy White and Saxophonist Mel Collins. The album was released in September with minimal commercial success.

Waters considered suing Wright, but ultimately decided an easier thing to do would be for Wright to leave the band at the end of the project. As the band was in financial trouble at the time, Wright agreed. Waters, Gilmour, Producer Bob Ezrin, Composer Michael Kamen and session player Fred Mandel played keyboards on The Wall. Wright was retained as a salaried session musician during concerts to promote that album in 1980–81, and became the only member of Pink Floyd to profit from the initial run of the costly Wall shows, since the net financial loss had to be borne by the remaining "full-time" members. Wright did not attend the 1982 premiere of the film version of Pink Floyd—The Wall. In 1983, Pink Floyd released The Final Cut, the only album from the band on which Wright does not appear. His absence from the credits was the first time fans realised he had left the group, which was officially confirmed some years later.

During 1984, Wright formed a new musical duo called Zee with Dave Harris (from the band Fashion). The pair had been introduced by a mutual friend, Saxophonist Raphael Ravenscroft. They signed a record deal with EMI Records and released only one album, Identity, which was a commercial and critical flop. Wright later referred to Zee as "an experiment best forgotten".

After Waters' departure in 1985, Wright began to contribute to Pink Floyd again, beginning with sessions for A Momentary Lapse of Reason. However, he did not legally rejoin as an equal to Gilmour and Mason, and was a salaried musician for the resulting tour, as his contract said he could not rejoin as a full member. On the album credits, his name was listed after Mason and Gilmour and his photo did not appear on the cover.

From the 1987 Momentary Lapse of Reason tour onwards, Wright and touring keyboardist Jon Carin favoured Kurzweil digital synthesizers, including the K2000 keyboard and K2000S rack module for reproducing piano and electric piano sounds. Wright retained the Hammond along with a Leslie speaker, playing it onstage and using it during the Division Bell sessions.

By 1994, he had rejoined the group full-time. "I am a full member," he clarified in 2000, "but contractually I am not on a level par with Dave and Nick." He cowrote five songs and sang lead on "Wearing the Inside Out" for the next Floyd album, The Division Bell. This was followed by the double live album and video release Pulse in 1995. Wright, like Mason, performed on every Floyd tour.

In 1996, inspired by his successful input into The Division Bell, Wright released his second solo album, Broken China, which had been co-written with lyricist Anthony Moore, who helped with production and engineering. The album covers the theme of depression and helped Wright come to terms with seeing friends affected by it. Musical contributions came from Pino Palladino on bass, Manu Katché on drums, Dominic Miller (known from his guitar work with Sting) and Tim Renwick, another Floyd associate, on electric guitar. Gilmour contributed a guitar part for "Breakthrough" but his performance didn't make the final mix of the album. Wright considered taking the album on tour, but concluded it wouldn't be financially viable. Sinéad O'Connor sang lead vocal on two tracks, "Reaching for the Rail" and "Breakthrough", with Wright covering the remainder.

In 1999, Pink Floyd touring keyboardist Jon Carin joined with Wright's wife to bring Wright and Waters back together after some 18 years apart; the two men met backstage after a tour date by Waters.

Wright played at several of Gilmour's solo shows in 2002, contributing keyboards and vocals, including his own composition "Breakthrough". In 2006, he became a regular member of Gilmour's solo touring band along with former Floyd sidemen Jon Carin, Dick Parry and Guy Pratt. He contributed keyboards and background vocals to Gilmour's solo album, On an Island, and performed live in Europe and North America that year. On stage with Gilmour he played keyboards, including a revival of the Farfisa for performing "Echoes". Wright sang lead on "Arnold Layne", which was released as a live single. He declined an offer to join Waters and Mason on The Dark Side of the Moon Live tour to spend more time working on a solo project.

On 2 July 2005, Wright, Gilmour and Mason were joined by Waters on stage for the first time since the Wall concerts for a short set at the Live 8 concert in London. This was the last time that all four post-Barrett Pink Floyd members performed together. Wright underwent eye surgery for cataracts in November 2005, preventing him attending Floyd's induction to the UK Music Hall of Fame.

In 2006, Wright joined Gilmour and Mason for the official screening of the P•U•L•S•E DVD. Inevitably, Live 8 surfaced as a subject in an interview. When asked about performing again, Wright replied he would be happy on stage anywhere. He explained that his plan was to "meander" along and play live whenever Gilmour required his services. The same year, he co-wrote the Helen Boulding B-side, "Hazel Eyes", with Chris Difford.

Wright's final vocal performance took place at the Syd Barrett tribute concert "Madcap's Last Laugh" at The Barbican in London on 10 May 2007. It was organised by Joe Boyd in the memory of Barrett, who had died the previous July. The first half featured a Roger Waters solo performance, while the second half concluded with Wright, alongside David Gilmour and Nick Mason, performing "Arnold Layne". His final live performance was as part of Gilmour's band at the premiere of Gilmour's concert DVD Remember That Night on 6 September 2007 at the Odeon Leicester Square, London. After an edited version of the film had been shown, the band took to the stage to jam.

Wright died at home in London of cancer on 15 September 2008, aged 65. At the time of his death, he had been working on a new solo album, which was thought to comprise a series of instrumental pieces.

Wright's last Floyd appearance, a posthumous one, was on 2014's The Endless River. The album contained mostly instrumental music recorded during the Division Bell sessions in 1993–94.