Step 1: Cut sides away from pit
Using a sharp paring knife, start just off the centerline and cut parallel to the seed along the long axis of the fruit. If you contact the seed, just curve away a bit to follow the contours of the pit. You'll leave more flesh on the pit, but it's not going anywhere. Once you get a feel for the seed sizes in your mangos of choice, this process will get even tidier. You should see the white pit showing in the center piece when you're finished.
Cut both sides off of the pit in this fashion, and set them aside.
Step 2: Remove skin from center piece
Now that you've skinned the center, go ahead and eat it. This is the best part of the mango, and it's your treat for cutting up the mango for your lazy friends or family. Use your teeth to scrape all of the mangoey goodness off of the pit, then remember to floss. Mango pits are fantastic, but also the best impetus to floss this side of corn on the cob.