Answer:
A basted egg is one cooked sunny-side up, but with the top cooked by "basting". It's often basted with hot fat from the pan, but some people do it with hot water instead. You cook the egg in a fair bit of fat, and spoon some of the hot fat over the egg while it cooks.
You can also get a "basted" egg just by covering the pan with a lid as it cooks, steaming the top to make sure it gets cooked. That's the way I do it.
The idea is that a plain sunny-side-up egg is often too runny on top, though many people like it that way, and an over-easy egg can over-cook the yolk if you like your yolk. You also risk breaking the yolk when you flip it. Basted eggs are best if you prefer an egg which isn't slimy on top but you still want a thick sauce of lightly-cooked but not solid yolk.