Look for mangos that have a bit of give to the flesh; you don't want any soft spots. They should also be pleasantly aromatic.
The first technique you can use is to slice off the two larger sides around the flat pit. These two pieces are called the cheeks.
Then, using a paring knife, slice into the flesh (not all the way through the skin) as if you were dicing. Turn the mango inside out and cut off the cubes.
You can also make vertical cuts and use a spoon to scoop out the mango in simple slices.
If you want mango slices or a finer dice, peel the mango as you would an apple.
Next, slice off the flesh from around the seed, in thin slabs. You can now julienne the mango slices, and chop into a small dice for a fine salsa, sauce or garnish.