Davison was born in
South Shields. started his career at
Seaham Colliery Welfare Red Star, before making two appearances for
Huddersfield Town between 1980 and 1981. He moved on to
Halifax Town on 1 August 1981 and holds a good record 29 goals in 63 appearances. This caught the eye of
Derby County and between 1982 and 1987 he made 206 appearances scoring 83 goals, contributing towards their promotion to the Second Division in 1986 and the First Division a year later. He played a total of 206 league games for the Rams, scoring 83 goals and became a Derby legend. Much to Derby fans' disappointment he was sold for £350,000 to a rebuilding
Leeds United squad, managed by
Billy Bremner, in November 1987.
Howard Wilkinson succeeded Bremner as Leeds managed in September 1988 and in the subsequent push for promotion Bobby became a cult hero for Leeds due to a high work rate and general rapport with the crowd. His scoring record of 35 times in 110 games is a good one for Leeds at that time. He helped them seal a return to the top flight as they sealed the Second Division title on the last day of the
1989–90 season by beating
AFC Bournemouth 1–0 at
Dean Court. However, Davison lost his first team place for the
1990–91 season following the arrival of
Lee Chapman, who was established as a top goalscorer. He played just five times in the 1990–91 First Division campaign and scored once as Leeds finished fourth. He managed just two league appearances in the
1991–92 campaign, which Leeds ended as champions. scoring once. ==Coaching career==