Brady was from a footballing family, with both his great uncle Frank Brady Sr. and older brother Ray Brady having been Irish internationals. His late older brother Frank won the FAI Cup with Shamrock Rovers in 1968 and made 2 appearances in the UEFA Cup Winners’ Cup, while another brother, Pat Brady, played with Millwall and Queens Park Rangers.

Brady started his career at Arsenal, moving to London to join the side on schoolboy forms in 1971, at the age of 15. He turned professional on his 17th birthday in 1973, and made his debut on 6 October 1973 against Birmingham City as a substitute for Jeff Blockley, and put in an assured performance. However his next match, in a North London derby against Tottenham Hotspur, Brady had a poor match, and Arsenal manager Bertie Mee decided from then on to use the young Irishman sparingly for the time being. Brady ended the 1973–74 season with 13 appearances (four of them as substitute) to his name.

Brady made his debut for Ireland on 30 October 1974, in a 3–0 win against the Soviet Union at Dalymount Park in a European Championship qualifier. Brady has claimed his favourite international goal was that against Brazil in 1987.

Brady was at the peak of his Arsenal form by now, as shown by one of his best goals for Arsenal; having dispossessed Peter Taylor he flighted a looped curled shot from the edge of the penalty area into the top corner, in a 5–0 win against Tottenham Hotspur on 23 December 1978. During this time he was voted the club’s player of the year three times, and chosen as the PFA Players’ Player of the Year in 1979. Being from Ireland, he was the first foreign player to win the award.

In 1974–75 Brady was a first-team regular at Arsenal, and shone as a rare light in a side that hovered close to relegation for a couple of seasons in the mid-1970s. With the appointment of Terry Neill as manager and the return of Don Howe as coach, Brady found his best form. His passing provided the ammunition for Arsenal’s front men such as Malcolm Macdonald and Frank Stapleton, and Arsenal reached three FA Cup finals in a row between 1978 and 1980. Arsenal won only the middle of the three, against Manchester United in the 1979 final, with Brady starting the move that ended in Alan Sunderland’s famous last-minute winner.

That season, Arsenal reached the Cup Winners’ Cup final (losing to Valencia on penalties), having beaten Juventus 2–1 over two legs in the semi-finals. Brady’s performance in the tie impressed the Italian giants and in the 1980 close season they signed him for just over £500,000, becoming the first foreign player to sign for the club since the Italian borders were re-opened for foreign transfers in 1980. He is remembered as one of Arsenal’s all-time greats, playing 307 matches for the Gunners, scoring 59 goals and setting up many more.

Brady spent two seasons with Juventus, wearing the number 10 shirt, and picking up two Italian Championship medals, in 1981 and 1982; Brady scored the only goal (a penalty) in the 1–0 win against Catanzaro that won the 1982 title. After the arrival of Michel Platini in summer 1982, Brady moved to Sampdoria, and went on to play for Internazionale (1984–1986) and Ascoli (1986–1987), before returning to London in March 1987, for a transfer fee of £100,000, to play for West Ham United, where he scored 10 goals in 119 games in all competitions. He was a member of the side relegated from the First Division in 1989 and played one season in the Second Division before finally retiring as a player in 1990. His last game came on 5 May 1990, a 4–0 home win against Wolverhampton Wanderers, a game in which he scored.

After retiring from playing in 1990, he managed Celtic between 1991 and 1993, and then Brighton & Hove Albion between 1993 and 1995. Neither spell was particularly successful, and at both clubs Brady’s tenure was overshadowed by the respective clubs’ financial problems. At Celtic, Brady failed to win a single trophy in his two-year tenure, and included a 5–2 defeat on aggregate by Neuchâtel Xamax in the 1991–92 UEFA Cup, one of the club’s worst European defeats in their history.

He rejoined Arsenal in July 1996, as Head of Youth Development and Academy Director. Although he was linked to the manager’s post after the departure of Bruce Rioch, Brady stated he was not interested in the role and Arsène Wenger eventually took up the post. Under Brady, Arsenal’s youth sides won the 1998 FA Premier Youth League, the FA Premier Academy League U17 title in 2000 and the FA Premier Academy League U19 title in 2002. Under his watch they also lifted the 2009 and 2010 FA Premier Academy League U18 titles, together with the FA Youth Cup in 2000, 2001 and 2009.

Brady first worked as a pundit for the BBC at the 1990 and 1994 World Cups before transferring over to RTÉ in 1998. Up until the end of UEFA Euro 2008, Brady appeared as a pundit on RTÉ Sport, along with Bill O’Herlihy, Johnny Giles and Eamon Dunphy. He contributed to RTÉ Sport’s coverage of the 2010 FIFA World Cup.

Brady was one of dozens of former managers linked to the Republic of Ireland manager’s job after the sacking of Steve Staunton in 2007. Instead, he became an assistant to new manager Giovanni Trapattoni alongside former Juventus teammate Marco Tardelli in 2008, while continuing to work as Director of the Arsenal Youth Academy. He stepped down from the Republic of Ireland post in April 2010 when his contract expired. He stated he would have gladly stayed on with Ireland were it not for his Arsenal commitments.

In September 2010, he returned as a regular pundit to the panel for Ireland’s UEFA Euro 2012 qualifiers and was part of the RTÉ panel for UEFA Euro 2012 and 2014 FIFA World Cup qualifiers. He was also part of RTÉ Sport’s studio coverage of the 2014 FIFA World Cup. He was also part of RTÉ Sport’s studio coverage of the UEFA Euro 2016 and 2018 FIFA World Cup.

On 30 January 2013, Arsenal announced that Brady would leave his role as Director of the Arsenal Youth Academy in May 2014.