How can I remove the peel and pit of an avocado without the whole thing turning into mush?

The California Avocado Commission recommends this (safe but wimpy - see below for a better way) three-step process:

1. Start with a ripe avocado and cut it lengthwise around the seed. Rotate the halves to separate.
2. Remove the seed by sliding the tip of a spoon gently underneath and lifting out. The other common seed-extraction method - striking the seed with a knife and twisting - requires some skill and is not recommended.

3. Peel the fruit by placing the cut side down and removing the skin with a knife or your fingers, starting at the small end. Or simply scoop out the avocado meat with a spoon. Be sure to sprinkle all cut surfaces with lemon or lime juice or white vinegar to prevent discoloration.

   

Source: California Avocado Commission

Notes:

• Another (and my favorite) way to remove the pit is by holding the half with the pit in one hand and striking the sharp edge of the knife used in step one into the pit, then twisting to remove. This is the "pro" way to do it, but takes a little practice. The video Nate referenced in the comments shows how to do it, it's not that hard.
• If you're going to be chopping the avocado up, you slice it up using the tip of the knife still in the shell, then scoop the sliced avocado meat out with a spoon.

After 4 years... the classical spoon based method for avocado processing is obsolete.

The Triptych Peel Method

This combines a well known method to remove the seed, with a scheme for conservatively reducing the skin tension by a series of shallow cuts along the surface. This process allows for direct removal of both seed and skin, with minimal effort and minimal wasted fruit.

• Get a ripe avocado and make sure your workspace is clear for cutting on. If there is a produce sticker, remove it. Gently cut through the skin until you find the seed.

• Rotate the avocado itself 360 degrees, firmly maintaining the knife against the seed.

• Take the avocado with both hands, and twist each half in opposite directions, to unlock one half from the seed. (Sorry for the blurry image, the camera rig got bumped during exposure)

• Gently press the knife into the seed, until you have gripped it well.

• Use the leverage of the knife to turn the seed until it unlocks. (Sometimes moving back and forth helps here.)

This concludes the standard seed removal, now for the triptych skin removal.

• Cut a small notch upwards, out of the avocado at the 1/3rd point.

• Cut another small notch upwards, out of the avocado at the 2/3rds point.

• Hold the avocado on your cutting surface so your notches are at the top, be sure your grip does not put your fingers in harms way! Now make two shallow scratches down the surface of the avocado, starting from each notch, just deep enough to split the skin.

• Take the skin by a corner and peel it off. Make sure to start on the side with no hole, to avoid the skin cracking.

• Enjoy! I have exaggerated the lines here as a visual aid, but the avocado half is still in one solid piece. Can be sliced into "rings" which are wonderfully convenient for sandwiches.

A related tip to prevent browning: Put lemon juice

• On the avocado that you are serving and
• On the half that you are putting into the fridge with plastic wrap.

The vitamin C (ascorbic acid) prevents the oxidation that turns the flesh brown.

UPDATE: The top answer on this question: Browning Avocados - What Helps? strongly suggests that my answer here is incorrect and just the propagation of a food urban legend.

1. Cut the avocado in half (around the pit)
2. Jab the pit with the pointy end of a knife
3. Twist the knife to dislodge the pit, hit it on the side of your sink until the pit flies off and hits you in the head
4. Use a big spoon to scoop the flesh out of the skin of the two halves.

You need to have a ripe fruit. It is ripe when the neck (narrow part) just gives under a light squeeze. You can tell a ripe fruit because the peel will come off in large pieces, not sticking to the fruit and tearing or coming off easily and crumbling. Take a knife with a sharp tip and cut through just the skin as if cutting the fruit in quarters lengthwise. Then, following the previous cutting of the skin only, cut the fruit in half to the seed. Then twist the two halves apart, the seed will stay in one half. Hack the knife into the seed embedded in the half avacado and twist the seed out. Then peel the two pieces of skin off of each half, if you have a ripe avacado, each 1/4 skin will come off as one piece. Enjoy.