Equity is the name of the
Actor's Union in the
UK (the equivalent of the
Screen Actors Guild in the
US). An Equity card is a very highly prized item as far as actors are concerned (required I believe for a speaking part in TV Drama etc.).
Amongst other things Equity are the people who ensure that no two actors in the UK use the same stage name. The name on your Equity card is the name that goes on any of your credits.