Condo is an abbreviation for condominium as John has said. Condos are owned, usually by the person or people living in it. A fee is paid each month for the maintenance of the building. An apartment can be owned, usually in much larger cities such as New York, but usually it is rented by the inhabitant, with the money paid to the owner.

condo is short for condominium condominium(condo) and apartment are similar. condo just sounds more expensive or is a higher class apartment. you might also come across the term "flat" which also means apartment.