The "elbow" is the trap, I assume.
If it is a PVC white plastic drain assembly, it is made to be just hand-tightened. There are beveled plastic washers that go with the slip nuts on the PVC pipe. The flat side of the washer goes toward the nut, and the beveled end goes toward the pipe. If the washer is reversed, it will leak.
If it is chrome metal pipe drain assembly, there are rubber washers (doesn't matter which way these are turned), but the nuts need to be tightened with a wrench or channel-locks.
If all else fails, smear a little clear silicone on the washers. Should seal the leaking, and it will come apart easily when you need to do it again.