Callaghan was born in Dublin and began his career with English club Stoke City progressing through the youth ranks at the Victoria Ground and signed a professional contract in 1984. Stoke were having a woeful 1984–85 season and with the team already relegated Callaghan was given the chance to play First Division football playing in five matches towards the end of the season. He began the 1985–86 season on loan at Crewe Alexandra where he made 10 appearances. On his return to Stoke he played in eight matches under the management of Mick Mills. He made just three appearances for Stoke in 1986–87 and was sold to Oldham Athletic in October 1986.
He was selected for the 1985 FIFA World Youth Championship finals in the Soviet Union but on 19 August, five days before it was scheduled to start, he was recalled by Stoke City as he was needed for first team duty by manager Mick Mills the following Saturday. However, Callaghan did not play and would not make a senior appearance for the club until six months later. The defender made two Republic of Ireland national under-21 football team appearances, playing alongside Denis Irwin and Pat Dolan in a 0–0 draw in Belgium in September 1986 and in the 2–1 defeat to Scotland at Oriel Park in October 1987.
He was also player-manager of Crusaders for one year and in 2000 the club qualified for Europe after he had made three European appearances as a player. Apart from a brief spell as Athlone Town manager in 2004, Aaron had been a coach at Longford Town from July 2002 until the end of the 2006 season. Following a year out from the game, Aaron was appointed Longford Town FC manager in December 2007. Due to financial constraints at the club Aaron resigned as manager on 30 August 2008. Aaron was one of the first coaches in Ireland to obtain the UEFA Pro Licence in 2007. He is currently a coach education tutor for the Football Association of Ireland and sports development officer for Dublin City Council. Aaron has played for the Republic of Ireland U21s and was appointed Rep of Ireland U.14 International Coach for 2 years in 1997.
Callaghan was appointed as Bohemians manager on 31 December 2011. Despite financial constraints at the club, he led his team to a comfortable midtable finish in a season that included a 4–0 victory over Shamrock Rovers and Bohs' first league win in Tallaght Stadium. He was rewarded for this good season by being offered a two-year contract extension by the Bohs board of directors, which he signed in November 2012. On 15 July 2013 Callaghan was relieved of his duties by the club's board of directors.