Free climbing: climbing using gear only for protection, just using your own body to actually make progress.

Aid climbing: climbing where using gear is one of the primary ways of making progress

French freeing is in-between, you're mostly free climbing but will occasionally aid climb to get past a section that's beyond your free climbing ability.

5.7, 5.9 are climbing grades in the Yosemite Decimal System - can be used to refer to an entire route, a section of a route, or even just an individual move.

Chimneys and offwidth (OW) are sizes of cracks. A chimney is big enough to get your whole body inside it. Offwidth is too small to fit your body, but too wide to jam a hand or a foot inside it (jamming is when you constrict a body part against both sides of a crack to create friction)

A cam is a piece of gear that expands to jam inside a crack. This is what you see the guy in the video placing. In this case he's talking about using a #6 size cam to help aid through an offwidth crack graded 5.9+