Depends on the airline and the route flown (time-wise). For example, LHR to SFO will normally be 3 crew members (typically a Captain and two FOs).
One of the FOs must be APIC qualified (Acting Pilot In Command) which involves training so that he is able to fully operate the controls from the left seat, and is familiar with the differences in flying from the left seat. The APIC qualified FO will occupy the Captain's position whilst the Captain is on his rest. All three crewmembers have a rest period at some point on such a flight.
On longer flights, like LHR to Hong Kong, two entire crews are carried (2 Captains and 2 First Officers). One is designated as the "operating" crew who are actually responsible for takeoff and landing, the other is the "relief" crew who take over during the middle of the cruise. So for a 12 hour flight the operating crew might do the first 3 and last 3 hours, whilst the relief crew does 6 hours in the cruise.
On the return trip, the duties are normally reversed and the guys who flew out as "operating" crew return to London as "relief" crew and vice-versa.
This is how it works in the UK anyway. Hope it was of interest. I believe a similar system works with longhaul US flights, although every country has its own reglations on flight crew duty times.
I used to love the smell of Jet-A in the morning...