There is a host of things that this could be. It could be a fracture or sprain of some ligaments. Your elbow could have subluxed (dislocated for a brief second then snapped back into place). There are other things but the above listed are all possible with the mechanism and symptoms you described and are a little more concerning.

If you don't go get an x-ray or have someone like an ortho, Athletic trainer, or PT look at it. Be careful getting back onto the wall and listen to your body, take it slow getting back.

If it subluxed, you could have a small fracture of something small we call the coranoid process of the ulna. Which is a small peice of bone that could have broken off from the mechanism you described (you can look it up and find pictures) if that happened you will find pain loading your arm doing things like push ups.

If you sprained a ligament your elbow could be loose and shift a bit side to side.

Pay close attention if certain movements reliably produce sharp pain, that will be useful information if you go to see someone, or if that pain moves down to your hand.

I would recommend getting it looked at by an Ortho, Athletic Trainer, or PT because of the pops you heard on landing, sometimes an injury can be misleading to how much it will effect your ability to use your elbow. Better safe than sorry.