OpenMP is a way to program on shared memory devices. This means that the parallelism occurs where every parallel thread has access to all of your data.
You can think of it as: parallelism can happen during execution of a specific for loop by splitting up the loop among the different threads.
MPI is a way to program on distributed memory devices. This means that the parallelism occurs where every parallel process is working in its own memory space in isolation from the others.
You can think of it as: every bit of code you’ve written is executed independently by every process. The parallelism occurs because you tell each process exactly which part of the global problem they should be working on based entirely on their process ID.
The way in which you write an OpenMP and MPI program, of course, is also very different.