I think a good way to define that expression would be "without result". When one's efforts "go in vain", it means that although they tried their hardest and gave their best effort towards accomplishing a task, no progress was made. When you do something in vain, whatever it is you did was pointless. Hope this helps :)

How about 'fruitless'? I've just checked the translation into Bulgarian and it seems to be the same idea. Does that help explain the concept?

What about "напразно"?

"In Vain" by Emily Dickinson gives two examples of use: If I can stop one heart from breaking, I shall not live in vain; If I can ease one life the aching, Or cool one pain, Or help one fainting robin Unto his nest again, I shall not live in vain. Aesop's fable "The Fox and the Grapes" gives another: A hungry fox saw some fine bunches of grapes hanging from a vine that was trained along a high trellis, and did his best to reach them by jumping as high as he could into the air. But it was all in vain, for they were just out of reach: so he gave up trying, and walked away with an air of dignity and unconcern, remarking, "I thought those grapes were ripe, but I see now they are quite sour."