Muay Thai elbow strike is an extremely effective close range attack. It is utilized to block the opponent's vision by cutting his eyebrow, so that blood may obscure it and if the cut is deep enough the fight may end by TKO. It is also effective in delivering a spectacular knockout by hitting his chin.
To be able to deal this kind of a deadly strike you must combine 3 factors at the same time: the speed - quickly twist your upper body in the same direction as an elbow strike, the power - narrow the angle of your strike to the maximum - and the precision- make an impact with the tip: the hardest and the sharpest part of your elbow.
This allows you to attack from different positions and to deliver an effective hit from different angles.
Elbow can also be used as a follow-up strike coming from the same arm; like hook-elbow or uppercut-levering elbow combinations. When used properly, it will be more damaging than a punch itself. Another interesting opportunity may arise when your block your attacker's hook or a swing: just step into his unprotected guard and spring up with an uppercut elbow blow right in his face.
Elbows are extremely tough and there is very little that can break. You can deliver all sorts of elbow strikes without worrying that this may deal a damage to your body. Obviously, you will need to wear an elbow protectors to practise elbow strikes and combos with a sparring partner.
With every elbow combination, you will have to close the distance first to get in range. You can do it with kicks and punches or even a knee strikes. Try and practise these combos:
left jab+right leg kick+left knee+right elbow
left hook+right uppercut+left elbow
It will not only protect you from getting hit but will also decrease an attacker's ability to strike again. Elbows are harder than fists so your opponent may hurt his knuckles trying to hit you. Obviously this only applies to a self-defense outside of the ring. It is good to toughen them up on a banana bag before you try it in a real fight.
You can - and should - use it in a combination. Block your opponent's punch with your elbow and then deliver a blow to his exposed face. Or hit him with it from a half-clinch position. You can also throw an elbow slash right after escaping from a clinch.
Experiment with it - then Practise it - and finally Implement it. There is no better weapon to target your opponent's face from a very short distance.
Also take a look at:
Muay Thai techniques
Muay Thai elbow