A Chairman is the head of a board of directors.
A Chief Executive Officer is responsible for the day-to-day running of a company.
The two positions can be merged into a single role - Executive Chairman - which, simplistically, provides one person with the power and influence to control pretty much everything associated with that company.
Markus Liebherr purchased Southampton Football Club (outright) in 2009 and, in doing so, became its owner. Markus subsequently appointed Nicola Cortese as Executive Chairman of the club. In doing so, Markus effectively handed over complete control of the running of his club to Cortese.
If recent rumours are to be believed, KL (who inherited ownership following her father's death) had become keen to remove some of that power from Cortese. This has been mooted as being one of the main reasons behind their disagreement.