I always consider this to be the partner technique of hiji atei. Although both are elbow techniques, this one is more useful because it can be used even when someone is holding you by the front of your clothing, whereas it can be hard to deliver a hiji ate in such a situation. "Mawashi" means "round", and although "empi" is taken to mean "elbow", it actually means "monkey's elbow". Go figure.
Target areas for this are most likely to be the same as haito: in other words, the side of the jaw, or the temples. If you are rising from the floor or are crouched down, mawashi empi has great potential when used to the side of an attacker's body, where it can be used to break ribs and incapacitate without threatening their life.
Start with your left elbow extended in front of you at shoulder-height, with your left fist against your right shoulder. Your right fist should be retracted to your side with your palm facing upwards.
Now swing your right elbow out at head height, moving it in an arc towards your opponent's jaw. Many beginners frequently move the fist of their attacking arm away from their body in an arc, as if delivering a hook punch - this is incorrect as it seriously compromises your technique, weakening the elbow strike. As you swing your elbow around, keep your right fist close to your chest. As your elbow reaches its target, rotate your wrist in a 180 turn so that your fist ends up palm downwards, with your knuckles pressed against your chest or shoulder depending upon your flexibility. The important thing is that your elbow swings sufficiently far round to hit your opponent. As you swing your attacking arm out, pull the other one back to your waist, rotating your fist so that your palm faces upwards.
Beginners commonly drive their attacking elbow forwards from the waist in a kind of shita tzuki strike, rather than moving it in an outwards arcing path. Although I'm not sure whether or not this weakens the strike, it certainly reduces its usefulness in situations where you are being held by an attacker - by the lapels for example. This is because the elbow cannot get past the attacker's arms. The correct technique is to raise the elbow out to the side so that your upper arm is horizontal, then swing the elbow round.