It depends. There are rotating exciters and static exciters with slip rings. There are also exciters with a static exciter that feeds a rotating transformer with rotating rectifiers and sometimes a circuit that shorts the rectifiers during start. In the latter case, the rotor winding is used as an asynchronous starting winding which is shorted by the thyristors that are part of the rotating rectifier.
So, there are a great many options. But most of them - I would say all - use thyristors to control excitation. The thyristors are phase controlled. A long delay before they are fired gives a low excitation and a short delay produces more voltage and hence more current in the excitation winding. The delay is counted in electric degrees. One cycle is 360 degrees so normal delay is between 30 and 150 degrees (with some exceptions).