What is elbow? If it's the external part (where the red circle is)as we can see in the attached picture then what is the name of the internal part against to the elbow (the angle that made by the flexion) But I think that I saw a label from the other side too.
The elbow is the part labeled in your picture.
The other side's medical name is "the antecubital space, the cubital fossa or the chelidon." according to that site I just found on Google, but I've never heard any of those terms.
In everyday English, it may be called the "elbow pit" (like armpit), the "crook of the arm" (usually used with the whole phrase, not just "crook" alone, and specifically used when one is carrying something there), or simply "the other side/inside of the elbow".