They're the beech blight aphid (Grylloprociphilus imbricator), also known as the boogie-woogie aphid. That dance is a defensive mechanism. When disturbed, the aphids raise their abdomens and wave their waxy white butt fluff in the air. The signal spreads through the colony, and soon the whole tree looks like it's covered in wriggling fur. Here's a closer look at what's going on:
The aphids feed on the sap of the beech tree. (If they're feeding when you disturb them, they'll stay in one place and keep drinking while they do their dance.) But even though they're stealing sap, these aphid colonies don't seem to do much lasting damage to the tree.
They will attract another noticeable adornment, though. A perfect contrast to the woolly white aphids, the sooty mold Scorias spongiosa feeds on aphid poop (also known as honeydew) and covers the tree branches in black fungus.
I'm fascinated by this interconnected chain: the fungus grows exclusively on the honeydew of this particular aphid, which feeds exclusively on beech trees. And this is only scratching the surface of the world that relies on beeches...Doug Tallamy
reports that beeches also support at least 100 other species of insects. But for now we'll just enjoy the boogie-woogie aphids!