It could be many things – muscle strain,tendinitis, ligament strain. Or you may just be moving your elbow in an unnatural way.
RICE it if it’s still sore.
Next time try pull ups rather than pull downs (i.e don’t use a machine or cable).
I find doing pull ups forces you to maintain much better posture and generally results in your form being much better than when using a machine (assuming you don’t start swinging around like mad). People using machines or cables often press the bar down in front of their body too much rather than rowing it into their sternum keeping the elbows back. This puts strain on the anterior shoulder and elbow. You can’t do this whilst performing a pull-up as your body weight won’t allow it.
Pull ups are also just much better for you as you will probably shift much more weight (body weight). It’s almost pointless doing heaps of reps of a light weight on a lat pull down. I’d rather do fewer reps with body weight.
Bent over row or pull up (wide pronated grip) are top exercises.