Tennis Elbow. Ice it regularly and wear a compression brace. Stop doing bicep curls until it heals up. The compression brace helps for sure when doing other exercises.
I know how it is, when something is injured you still want to keep doing the same exercises. That's the time to try something new though. There are lots of other exercises besides curls you can do at the gym. Stop with the curls for a while.
Tennis elbow is a common injury from overuse and hyperextension. That is what the doc will likely diagnose too.
It will get better and stronger, just takes a couple months to heal sometimes which sucks.