If it is herbeceous, it is supposed to die back to the ground every year.

Even if it is a tree peony, if it has enough root mass, it should still be able to resprout an may flower again this year if you accidentally cut it down.

Thank you! Nope, it isn't a tree! Generally do people cut them down in the fall or let them naturally die back? I'm wondering if my neighbor who shares the fence that its along was aware of my ignorance and helped me out by snipping it!

For herbeceous peonies, I generally cut them back once they start dying back in the fall or early winter because you want to make sure the dead canes don't shade out the new sprouts.

Now for tree peonies, I would never cut them back because they store a lot of their energy in the above ground growth, and they are shrubs. If you cut them back, they will likely come back again, but the overall production may be reduced and you may not get blooms for that year as the underground shoots may not be flowering buds. It is OK to bury tree peonies deeper or lay them on their sides to encourage more sprouts.