If we're talking about two rigs that have similar configurations, one of the most useful copy\paste methods is by using the timeslider.
Select bones on animated character (make sure all selected bones are keyed) Selection order is important, as you will have to select the corresponding bones in the same order on your target character.
Shift\Drag in your timeslider to create a red selection range. Right-click "Copy" in the timeslider range.
Select bones on target character (once again, make sure you do this in the same order as when you selected bones on your animated character.)
Right-click in timeslider- "paste\paste"
If this isn't the case, I can describe a method for constrain\relative retargeting that's a bit more setup-intensive but may provide greater flexibility and control.