Born in Bellshill, Lanarkshire, Black spent his early life in the Glasgow area before his family moved north to Nigg, Highland due to his father's job in the offshore oil industry, and he signed for Aberdeen in 1980 (a week after Bryan Gunn who came from the same part of the country). Black was heavily involved in the most successful era of Aberdeen's history under the management of Alex Ferguson, the pinnacle being their victory in the European Cup Winners' Cup in 1983 with a 2–1 win over Real Madrid, in which the then 19-year-old Black scored the first goal. Black's main assets were his strength in the air (despite not being particularly tall) and goalscoring instincts. In February 1983, Black scored a hat-trick in a 3–1 win against Celtic, as of 2020 still the last visiting player to score three at Celtic Park. He made 180 appearances (30 as substitute) in all competitions for Aberdeen and scored 70 goals between 1981 and 1986, winning two Scottish Football League titles, three Scottish Cups and a Scottish League Cup as well the Cup Winners' Cup and the European Super Cup in 1983. He missed out on a potential fourth Scottish Cup medal when he was overlooked for the 1986 final having already agreed to move abroad.