A voltage source is "heavily loaded" when something is drawing a large amount of current from it. "Large" in this context is relative to what the voltage source can deliver, or perhaps relative to some normal or typical operation within a system.
For example, it might be reasonable to say a 12 V 10 A supply is heavily loaded when something is drawing 9 A from it. It wouldn't make sense to say that when the load is only 1 A.
A voltage source being heavily loaded does not necessarily imply that the voltage is sagging, although that could be a biproduct of some types of supplies being heavily loaded. A battery, for example, would be expected to have at least somewhat lower voltage when heavily loaded than when lightly loaded.
Most voltage supplies nowadays are regulated, so will keep the voltage reasonably constant over the whole valid range of output current. If such a supply were loaded to the point that voltage dropped significantly, then it would be "overloaded", not just "heavily loaded".
In the end though, "heavily loaded" is a qualitative term at best, with no exact meaning. If you really want to know how heavily or not a supply is loaded, get the current rating of the supply and the actual load current.