This seems to come from the same root as the Latin "albus", which means white. Is this so?
It originally comes from Lat. ”alveus” which means ”cavity” or "beehive". As time has passed the meaning has somehow wandered away to refer to the bees themselves. wiktionary.org is a great friend :)
It seems that Latin apis (Italian ape) or Latin apicula (Spanish abeja) does not have a descendant in Romanian.
We do have "apicultor" = bee keeper