Elesh Norn’s triggered ability will trigger separately for each permanent dealt damage by a source an opponent controls, plus once if you’re dealt damage by that source. For example, if an opponent casts a spell that deals damage to you and two creatures you control, Elesh Norn’s ability will trigger three times. Each of those abilities will cause that opponent to lose 2 life unless they pay {1}.

Similarly, the triggered ability will trigger separately for each source an opponent controls that deals damage to you or a permanent you control. For example, if three creatures controlled by an opponent block and deal damage to an attacking creature you control, Elesh Norn’s ability will trigger three times.