A trial by combat is essentially requesting to be judged by the gods. Denying someone the request of a trial by combat is possible, but has severe political consequences, since it implies that the denier doubts the judgement of the gods. However, it seems that only highborns are permitted to request a trial by combat.
A Trial by Combat:
- May be requested at any point before or during a trial.
- May be advised against, but even sovereign Lords and members of the Royal Family rarely even consider denying such a request outright.
- May be fought by the parties themselves, but it is not unusual for either or both parties to choose champions to fight in their stead.
- Ends when either party yields or is killed; or when the accuser takes back his accusation; or (presumably) when the accused declares himself guilty.
Considering the religious aspect of the trial, the champion of the just cause should prevail. Denying a champion is declaring that person to be above the gods.
It is less grievous to deny a choice of champion if, say, the champion is unable to arrive for the trial in a reasonable amount of time, or is unfit to fight, especially if there are other champions more readily available to the accused. If the accused is a royal, the champion must be one of the Kingsguard.
This is prior to the events of the books/show:
Of course, some find loopholes. For example, Mad King Aerys proclaimed fire to be his champion when Ned Stark's father and older brother went to King's Landing to seek justice for the kidnapping of Lyanna Stark.