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. These are the easy ways to find them (else than finding them by just digging rock)
Surface Cave Entrance
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 entrance for a cavern.
Common Mob(s) in the cavern (Spiders as this picture shows)
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
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.
Example of a Gravel vein to a entrance into a cave.
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.
(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
- 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.
Using A Compass
If you right click with a compass when looking at a block, then you can move through it. Making it easy to find caves as all you have to do is look down and click. The best place to use this is at the side of a cliff/mountain.
F3 Key
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! Courtesy of Rabid_frog (MC Name)
X-ray texture packs
All texture packs which edit an opaque block so that a part of it/ all of it is transparent can help find caves. When the block is looked through, it shows every air pocket above and underground. However, this may be considered as cheating.
An album of images can be found here :http://imageshack.us/g/62/20110709113322.png/ The example see-through block is white wool.