Mangoes are similar to peaches. They should be very sweet and juicy. Low-quality (but not rotten or unripe) mangoes are fibrous and have a turpentine smell.
Mangoes do not keep well. Non-rotten fresh ripe mangoes are very rare. Many mangoes start rotting right on the tree. The sliced mango in the picture has in fact just started to rot, as you can see by the dark-appearing translucent area. This mango would have a bit of a funny smell, vaguely like plastic or some sort of cleaning fluid. The translucent area is starting to liquefy. An unripe mango is crunchy, pale yellow on the inside, and not very sweet. Mangoes should not have dark spots on the side; these indicate that the mango has started to rot. Mangoes generally rot from the non-stem end, from the seed or pit, and from any dark spots on the outside. Black fibers indicate a different sort of rot, also bad.
When purchasing a mango, obviously avoid any signs of rot. A mango should feel solid, but not really hard. A mango should have a noticeable good smell. The small elongated variety with a pointy end will generally be of better quality than the larger and more rounded type.
There are many ingenious ways to open a mango, most failing to let you examine the mango for bad spots. (you can take off the sides and then flip them inside out for example) The more-or-less obvious method is best. Slice off the stem end with a sharp knife. Next, use the knife to help you peel the mango. Dig the knife in a bit at the stem-end cut, creating a pull tab. Peel away from this tab. Repeat all the way around the stem end, then elsewhere as needed. Be gentle in holding the fruit as you do this. Next, notice the symmetry. A mango contains a large flat seed that divides the fruit into two halves. Slide the knife along each side of the seed, cutting the fruit into three pieces. Cut away and discard any translucent areas. If you had to discard anything, rinse the remaining part to get rid of juices from the rotten part. Chop up the fruit as desired. The middle section may give you trouble; if it is good then you might just gnaw it from the seed.
Cooked mangoes lose the characteristic mango smell and flavor. This can be desirable if you want a mush that tastes like sweet potatoes. You could serve cooked mangoes with meat.
Green mangoes are sometimes used in cooking. The whole mango may be chopped or thinly sliced, including the skin and seed.
Not many people are allergic to mango sap, but as a precaution, have an Epi-pen ready. Drops of sap may be found near the stem end of a mango. Mangoes are related to papayas and cashews.