There are 5 HMs in the game. You get them by talking to other people. They're never found just lying on the ground.

You can use a HM to teach a Pokémon a special ability. Unlike TMs, these can be used as many times as you want, but the moves can't be forgotten later.

The 5 HMs are:
Cut - Cut down trees that are in the way.
Flash - Light up dark places.
Fly - Instantaneous transportation to any town you've previously visited.
Strength - Move rocks.
Surf - Swim in the water.

HMs can also be used in battle. Most are very powerful moves.
Cut - a Normal attack. Causes damage.
Flash - Normal. Decreases the enemy's accuracy.
Fly - a Flying-type attack. The Pokémon that uses it flies up into the air and makes the enemy's next attack miss. Then it causes a large amount of damage.
Strength - a Fighting move. Causes damage.
Surf - a Water attack. Causes damage.

Where are they?
Cut is on the S.S. Anne. Talk to the captain to get it.
Get Flash from Professor Oak's Aide, in the building near the Cerulean City Diglett's Cave Entrance after you've captured 15 kinds of Pokémon.
Fly is in the house that is north of the Celadon City entrance to the Bike Trail. Does this make any sense? I didn't think so. Try this: Cut down the bush to the west of Celadon City. Go north and west, and into the house. Talk to the girl and you'll get Fly.
The Game Warden will give you Strength after you return his gold teeth.
If you find the Secret House in the Safari Zone, you'll get Surf.