I would define it as energy (usually heat energy) being transferred from one place to another by a moving fluid.
For example when a hot object cools "by convection" in a fluid, the small amount of fluid actually in contact with the object is warmed by conduction (i.e. the velocity if the fluid molecules is increased by interaction with the faster moving molecules in the hot object).
The density of the hot fluid is reduced, so it moves (upwards) and is replaced by colder fluid. It's the movement of the fluid which causes a relatively high rate of energy transfer from the object to the fluid.
"Forced convection" involves making the the fluid move by some other method - e.g. a fan heater blows a stream of air over a hot object to increase the rate of energy transfer, compared with just letting the hot air circulate on its own.