A cave is a place where you can find many resources, or it's just a little room. Sometimes you find them on the surface, others are buried underground. Some caves connect from the surface all the way to bedrock. These are the easy ways to find them (else than finding them by just digging rock). A tutorial on most of these, can be found here.
Surface Cave Entrance
Common entrance for a cavern.
Usually, when you go exploring the surface, you might find cave entrances. Watch out though, mobs could be lurking at the entrance. This method is known as Cherrypicking, but you can also use a good strategy called strip mining.
Common Mob(s) in the cavern (Spiders as this picture shows)
Ravine Entrance
Ravines commonly intersect caves as well as abandoned mineshafts. Resources are a plenty, but be wary of flowing lava. Don't worry too much about mobs unless you find a spawner or find an underground ravine.
Resource veins
As caves have a lot of resources, you can enter a cave by digging resource veins. When digging veins you may hear mob sounds which means there is a cave nearby. Be careful though or you could come face to face with a creeper!
Dirt/Gravel veins
Example of a Gravel vein to an entrance into a cave.
Caves have dirt/gravel veins on them depending on how big they are. A big cave has many dirt/gravel veins and a short cave might have only one. Even if you find no cave by digging these, you will have created a cave-like room. Remember: If you find a dirt vein near the surface, don't dig it because you will end up in the surface. It is more suggested to dig a dirt vein if it's 15 blocks below the sea level. It is highly likely that you will find rare or semi-rare materials buried in or around dirt and gravel veins. Be sure to fully excavate these veins but exercise extreme caution as well, you may discover dark caves, mobs, lava etc.
Dungeons
Dungeons are places where monsters generate and there is loot in chests inside of them. Most dungeons are connected with caves. However, dungeons are quite rare and are not an easy way to find caves as most are found in a pre-existing cave and very few are found on the surface. When you find a dungeon, there will most likely be a monster spawner. You can disable the spawner by placing torches on and around it.
(More detail on Dungeons can be found here)
Sounds
Another way to find caves is to follow sounds from under the ground, or behind the walls in case you are already in a cave and trying to find another one. Practically, any sound you hear indicates an open area in the direction of the sound. This includes:
- Ambient sounds
- Lava noises
- Monsters' noises
- Footsteps
- Flowing water
- Fire (comes from Abandoned Mineshafts' wooden structures, which occasionally catch fire from lava)
Determining the exact direction of the sound is tricky. Few hints:
- If the sound does repeat time to time, try turning your head (ingame) while listening. First make it sound in one of your ears (in headphones/speakers) louder than in other, that would give you general direction. Second, try finding the exact direction by turning your head (ingame) so that its volume is as even as possible in both of your ears (in headphones/speakers).
- It is usually unclear if the sound comes from below, or from above. Try digging two blocks down, and climbing two blocks up. Compare the volume. The closer you are to the source of the sound, the louder it gets. If it is still the same, then the source is probably on your original level.
- If the sound does not repeat, note that you can hear sounds generated from no further than 16 blocks away (for most of the sound sources). That means you can dig no further than 16 blocks away to find another cave.
Also, a sound like the howling wind can be heard when the player is in close proximity to a cave or other opening. If you seal up your basement you'll hear this sound as well. The sound is created by any cavern in which you are within 3 blocks of, and it is incredibly hard to pinpoint. The easiest way to find the caverns is to simply smash through walls.
F3 Key (Before 1.8)
Pre 1.8, you could press the F3 key to show a mob's ID. This would show the location of a mob in a cave (when not on Peaceful); if you dug towards the mob, you would have most likely revealed a tunnel system that the mob was in.
Pistons
This is a simple and effective technique. Simply place down pistons facing the wall(s) you are looking for caves behind and power it with a redstone torch. If there is an empty space within 12 blocks the piston will push. Simply dig in that direction and you should find a cave!
Sandpits in deserts
Sometimes, it is possible to find a cave when wandering through the desert. To do this walk around until you find an odd 2 or 3 block drop in a small location, that looks out of place. Dig stairs into it and you might find a cave.
Transparent blocks
This is a very good a and simple technique to help you find caves all you need to do is make a 2 long, 1 wide and 2 high space and place a piston (doesn't have to be sticky) in front of you against the wall with a transparent block in front of it and then all you have to do is power it by any means necessary (it works better at a farther render distance) and it should show you cave systems in your area and mobs in them.
An album of images can be found here :http://imageshack.us/g/62/20110709113322.png/ The example see-through block is white wool. You can also mine and find Caves.