He spent a large portion of his career at Aberdeen over two spells, and is ranked 11th on the club's all-time appearances list; he has been described as "arguably the club's last great player".
Born in the village of Portsoy in Aberdeenshire, Jess began his career in Glasgow as a trainee striker at Rangers (alongside future Scotland teammate John Spencer) but was allowed to leave in 1987; he soon moved back to his home region, signing for Aberdeen. Having made his debut at the end of the 1988–89 season, 18-year-old Jess made an impact in the first team from the outset of the following campaign, starting against Rapid Vienna in the UEFA Cup and against Rangers in the 1989 Scottish League Cup Final at Hampden Park, which Aberdeen won 2–1. The club finished runners-up in the Premier Division, and Jess picked up another winner's medal as an unused substitute in the 1990 Scottish Cup Final.
In 1990–91 he scored 13 league goals and linked up with Hans Gillhaus to great effect as Aberdeen missed out on the title on the final day. The strikes including a hat-trick away to Dundee United, four away to Dunfermline and three across two home wins over Celtic. He won the PFA Scotland Young Player of the Year award for the season. Although Jess played 42 games in 1991–92, the season was a disappointment as Aberdeen finished only 6th in the league. He scored in a September victory over Rangers at Ibrox Stadium which proved to be Aberdeen's last win at that venue until 2017.
1992–93 brought an improvement as the club were league runners-up and reached both cup finals. Jess scored the winning goal in the semi-final of the 1992–93 Scottish League Cup against Celtic and played all 120 minutes of the final which ended in a 2–1 defeat to Rangers. In March 1993 he suffered a fractured ankle in the quarter-final of the 1992-93 Scottish Cup against Clydebank but was rushed back to fitness after only two months out and appeared as a substitute in the final of the competition, which Aberdeen again lost 2–1 to Rangers. Despite failing to win any trophies and missing part of the season with his injury, he won the PFA Young Player award for a second time.
In 1993–94 Jess helped Aberdeen to another second-place finish in the Premier Division. He also scored five goals in four matches in the Cup Winners' Cup, finishing joint-top scorer in the competition. The next season almost ended in relegation for the club and Jess started only 15 league games due to a persistent foot injury. He returned as a regular starter in 1995–96 playing in a more withdrawn central midfield role to accommodate strikers Dodds, Booth and Shearer. He produced an impressive performance in the semi-final of the League Cup as Aberdeen overcame Rangers, scored a long-range goal in a league game against the same opponents a few weeks later and was in the team which won the 1995 Scottish League Cup Final, beating Dundee.
Following this victory, he decided to seek a new challenge in his career and requested a move with his contract due to expire. As he had impressed in Europe against Italian opposition (Torino), Serie A clubs showed an interest and Jess has stated that Sampdoria were keen to sign him under the Bosman ruling at the end of the season, but he was determined that Aberdeen should receive a transfer fee, therefore agreed to a £2 million transfer to Coventry City of the English Premier League in February 1996. He left Aberdeen for the first time having made 253 official appearances for the club, scoring 63 goals, and the transfer fee they received for him was a record amount.
25-year-old Jess made his Coventry debut against Middlesbrough on 24 February 1996 and made 12 league appearances in the latter part of that season under manager Ron Atkinson, scoring what would prove to be his only goal for the club against QPR. In 1996–97 he played 27 times in the league but only started 19 of those, and scored an own goal in a game against Manchester United. He scored twice in Coventry's run in the 1996–97 FA Cup, netting in the third round game against Woking, and again in the fourth round victory at Blackburn Rovers. By that time Gordon Strachan had taken over from Atkinson as manager, and in the close season he allowed Jess to leave as he could not guarantee his place in the team. Despite his stay at Highfield Road being fairly brief, he later described the experience as "fantastic" due to the excitement of the games fighting relegation.
In summer 1997 Jess returned to Aberdeen, rejoining Roy Aitken's side for a fee of £700,000. This spell at Pittodrie was shorter and less successful than his first, although he did manage a career-best total of 14 league goals in 1998–99 including on the opening day which was the first goal of the Scottish Premier League era. He also played in the 2000 Scottish League Cup Final and 2000 Scottish Cup Final, both of which ended in defeat.
Having been a regular starter since his return he played less frequently in his final season, 2000–01, under manager Ebbe Skovdahl. In December 2000 he stated publicly that he may not sign a new contract and suggested that Aberdeen's ambition did not match his own (unaware of the club's financial plight at the time), alienating him from the hierarchy; he soon departed for a second time upon the opening of the winter transfer window, having made a further 126 Aberdeen appearances (31 goals).
Jess returned to the Premier League at the age of 30 with Bradford City, initially on loan, and scored on his debut for the Bantams on 1 January 2001 against Leicester City. Bradford were relegated in last place but the loan move was a success (17 games, 3 goals), with the deal being made permanent in the summer when Jess's Aberdeen contract expired. In the 2001–02 First Division he made 45 league appearances and finished the season as top scorer with 14 goals (including a hat-trick in a home win over Watford), but financial problems meant he was released by Bradford in June 2002 following the collapse of ITV Digital.
He joined Nottingham Forest in August 2002 and in his first season steered Forest into the play-offs in 6th place. However they dropped to 14th the next year and were relegated in 2005, and Jess moved on as a free agent.
In July 2005 he signed for Northampton Town and in his first season with the club achieved promotion from the fourth tier; in April 2007 he was released from his contract with The Cobblers, aged 36, and retired from playing.
He later rejoined Nottingham Forest as a coach for their youth team from 2007 until he left the post in May 2012. He also spent time on the coaching staff at Peterborough United, joining in August 2012 to work under Darren Ferguson as a replacement for Mark Robson but departing less than a year later, replaced by Gary Breen (coincidentally, once a teammate at Coventry for a short period).
Having previously been selected for the Under-21 side, and after declining an approach from Northern Ireland (his father's birthplace), Jess made his debut for Scotland in November 1992 against Italy. He went on to earn 18 full caps. As an intermittent member of the main squad, he also appeared twice for the B team in 1994 and 1996.
Whilst at Coventry City he was selected for UEFA Euro 1996, and made one appearance during the tournament as a substitute in the 0-2 defeat to England. His two international goals were both scored while playing for Aberdeen, although four years apart. The former goal came against San Marino in 1995, and the latter was scored in a 1-2 UEFA Euro 2000 qualifier defeat at home to the Czech Republic in March 1999; he appeared from the bench in the return fixture three months later, which proved to be his final Scotland match.
In April 2009, at the age of 38, Jess suffered a mild stroke. During further tests it was discovered that a hole had been present in his heart all his life (heart scans were not part of the medical examination during his transfers, and may have jeopardised his career had they been) and in 2010 he underwent surgery to correct this.
In 2015 Jess was voted into Aberdeen's 'greatest ever team' by supporters of the club.
In 2016 he was living in Barcelona.