There are two main types of green mango: ripe and unripe. Unripe green mangoes can be found anywhere ripe mangoes can be found, only earlier in time. Unripe green mangoes can be very very sour indeed, and cause your mouth to pucker uncontrollably, so I avoid them. Ripe green mangoes are a different kettle of fish altogether. As far as I know, they only grow in Thailand, but they are shipped around the world and I can usually buy them here in Toronto. Look for them: they're delicious.
Thai green mangoes have very dark green skins and are quite large and sort of eye-shaped. They can weigh up to 1 lb (about 2 kg). Although they are dark green on the outside, inside they can range from a pale green to a bright orange. Sometimes the skins have a kind of black scab on them. It looks yucky, but don't worry: it's just sap that has leaked out and dried. You peel and discard the skin anyway.
For a long time I was baffled by these mangoes because I couldn't tell from the outside what the insides would be like. Both are nice, but the pale green flesh is sour-sweet, like a tart granny smith apple, while the orange flesh is very sweet, like a regular ripe mango, only firm, not mushy. I use these mangoes for green mango salad, and I prefer the sweeter, oranger mangoes. I asked the people in the tropical fruit store where I buy them, and they said the orange-fleshed ones are softer. However, when I squeezed the mangoes, they all felt pretty hard to me, so I didn't believe those people, and ignored their advice. For a year or two I played mango roulette. Finally I decided that those people sell fruit, so they probably know what they're talking about, and I should listen. And indeed, the orange-fleshed mangoes are almost imperceptibly softer.
Want to see what they look like? You can find pictures on the web at: http://www.dpi.qld.gov.au/horticulture/5332.html.