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I was watching a woodworking YouTube video which showed a metal insert being permanently hammered into a pre-drilled hole at the bottom of a table leg. A large machine screw with a padded head was then threaded through an opening in the insert and into the pre-drilled hole. The result provided an adjustable foot for the leg which could extend the length of the total leg by up to about an inch.

Here is a bad drawing of what the insert looked like in the video...

My question is what are these inserts called? I want to order some but don't know what to look for.

The ones in your sketch are T-nuts, and are typically hammered into the wood. The prongs bite into the wood, keeping it from moving, and the inside of the barrel is threaded to accept a bolt.

Another commonly-used piece of hardware that performs a similar function is a threaded insert. These have threads on the outside of the barrel as well, and are screwed into a hole in the wood. The threads on the outside bite into the wood, keeping it from moving, although some people will also put a small amount of epoxy in there, too.

From what I can see, there's not a whole lot different about them other than the size. T-nuts require a bit more wood around the hole to provide a place for the prongs to bite into. Thus, threaded inserts can be used on smaller pieces of wood than T-nuts (e.g. end or edge grain of 3/4" or 4/4 material). Threaded inserts seem to be more common in knockdown furniture whereas T-nuts seem to be more common in "mechanical" builds (knobs, jigs, etc), although that's just based on me watching woodworking videos on youtube and nothing concrete.

Links are for example only, and are not an endorsement or recommendation.