At the utility company's transformer there are two windings. The primary connects to the high voltage power lines running from God knows where. The secondary winding reacts to the magnetic field produced in the primary, it produces a voltage of 240 when measured at both ends of the winding. Because someone famous 130 years ago decided a lower voltage might be useful, the utility transformer secondary is center tapped. As we now have 3 conductors available to run to the house, (two ends and center) from the transformer secondary, (120v from either end of winding to center, or 240v end to end). For a number of reasons it is important to keep all current flow within these three conductors, the center tap is connected to earth at the transformer, its conductor is connected to earth at the house. This center tap connector we call grounded or neutral, is usually identified with white, it only conducts the difference in current between the two "hot" conductors leading from the transformer secondary winding.
At the house you should have two heavy conductors connected to a service disconnect or breaker, this in turn feeds two bus bars like interlocking fingers upon which branch circuit breakers latch. It now sounds like you have 240/120 volts running to your garage. I sense there are code issues, but you may qualify for the "grandfather" exemption. Your two insulated conductors to the garage should connect at house end to a double pole breaker. Your uninsulated conductor (I would tape up each end with white electrical tape) connects to the neutral buss in the house and garage. Any conductors (water pipe etc.) between them disregard this. You need driven ground at garage. Bond this to neutral.