Okay, as hard as it is to describe a complex physical process, I'm gonna wade in and try.
Ideally you will use a sharp knife at least 6 inches long in the blade. Observe that the mango is shaped a bit like an egg that has been pressed semi-flat. The stone has a very similar shape and takes up a heartbreaking amount of space in the middle of the fruit.
Hold the fruit upright, small tip on your cutting board, the stem location up. The pit runs the same way the fruit does, so hold the fruit very firmly with your fingertips at about the north pole and rotated so that the narrowest edge of the fruit is pointed towards you.
Begin to cut down gently about 1/2 away from the stem. You are likely to graze the edge of the stone, just angle out very slightly and shave around it but continue to cut downward. You should end up with an oval of fruit a bit smaller than your cupped hand. Now turn the fruit around and do it again. You should have two ovals of fruit with skin on and a stone with a thin band of fruit 'round it's edges but nothing much to the front or back.
If you like, you may switch to a paring knife at this point.
Trim the strip of fruit off the stone by cutting straight thru the fruit towards the stone and then turning and the fruit towards the blade, a bit like peeling an apple. Lay that piece of fruit peel down on the board. Anchor one end of the strip firmly with the index finger of your off hand. Put the knife blade just in front of your finger, pointed away from you, blade parallel to the board and very flat. Gently cut the fruit from the peel.
Now for the two ovals of fruit. Use the tip of your knife. Put the fruit, skin side down on the board and gently cut throught the fruit but not the skin in three or four parallel lines, about 1/4 inch apart. Now pivot the fruit to to one side and do that again, so you have cut the fruit into 12 or so little cubes that are still attached to the skin.
If all you want to do is eat the fruit, pick the piece up and push the skin from the back. Eat the fruit like a combo of eating a peach whole and eating corn off the cob. If you want the flesh for a recipe, slice the piece in half along one of the lines that you have laready 'perferated'. The quartered fruit should be easy anough to hold flat, skin side down on your board. Cut the cubed fruit from the skin the same way you cut the strip of fruit loose before.
Man, did that make ANY sense?