An ackee is a kind of fruit (Blighia Sapia). Well, biologically so: it grows on a tree. Culinarily, it's treated as a vegetable. I've never had one, but I'm told it tastes rather like scrambled eggs. Only eat ripe ones, and only the right parts, or you can get Ackee Poisoning. It is often eaten with salted fish, though (ackee and saltfish being a signature national dish of Jamaica).
Also spelled "ackie" and several other variants.