A domestique is, literally, a servant.
On a cycling team, it's his or her job to help the team leader win the race. That often means riding for long stretches in front of the leader, making a slipstream that allows him or her to conserve energy for a breakaway or climb. It can also include inserting themselves into a breakaway and forcing other teams to chase, or to chase another breakway that threatens their team.
A domestique can also bring water or food from team cars to the leader. He or she might also be called upon to give up a wheel--or even a bicycle--if the leader's machine fails. I've also heard of domestiques giving leaders the jerseys off their backs!
How far should a domestique go to help the team's leader win?