A chairman is the person that runs a football club and is the boss of the club's manager. The chairman provides the manager with money for transfers, assesses the manager's performance and makes higher-level decisions about the club, such as whether to expand the stadium or improve the training facilities. On rare occasions a chairman can also decided to sign a player without the manager's permission - this is only done by certain chairmen.
Outside of the UK, a chairman is often referred to as a club president.
In Football Manager, the chairman of the club can leave if he gets very old or doesn't want to be there anymore, whatever reason. During that time, there will eventually be a takeover and a new chairman assume control with his ideas and his money. The current manager could be sacked when a new chairman is in place. The chairman profile is always hidden, but can be seen in the Data Editor and FMRTE.