How do you remove the pit from a mango?
Hold the mango upright - so the place where the stem was is at the top. 
Turn the mango so the widest side is facing you. 
The pit mimics this same shape - it's tall, wide, and not very deep. 
Cut all the way around as though you are creating two halves, one whole piece facing you, and one facing opposite. Go ahead and peel this side (using a vegetable peeler).
Slide your sharp paring knife between the peeled flesh and the pit. As you separate a section off the pit, slice it away from the rest of the flesh. If you don't use a sufficiently sharp knife, you just end up with a mess. 
Peel the other side and slice it the same way.


The reason I peel only half at a time is so I have something (the remaining peel) to grip as I slice the mango.