Cut away sides from pit: The mango has a flat-ish oblong pit in the center of it. Your objective is to cut along the sides of the pit, separating the flesh from the pit.
Holding the mango with one hand, stand it on its end, stem side down. Standing up the mango up like this you should be able to imagine the alignment of the flat, oval pit inside of it. With a sharp knife in your other hand, cut from the top of the mango, down one side of the pit.
Then repeat with the other side. You should end up with three pieces: two halves, and a middle section that includes the pit.
Make crosswise cuts in flesh: Take a mango half and use a knife to make lengthwise and crosswise cuts in it, but try not to cut through the peel. Invert the mango half so that the cut segments are sticking out like a hedgehog.
Cut or peel segments away: At this point you may be able to peel the segments right off of the peel with your fingers. Or, you can use a small paring knife to cut away the pieces from the peel.
Cut away pit: Take the mango piece with the pit, lay it flat on the cutting board. Use a paring knife to cut out the pit and remove the peel. You may be able to extract a little extra mango from around the pit.