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. When the cut is deep enough it may result in TKO.
It is also very effective for KO-ing opponents cold, especially a spinning back elbow.
To connect an elbow strike you have to combine 3 different factors at the same time:
Different elbow strikes allows you to attack from different positions and to deliver an effective hit from various 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 deal all sorts of elbow strikes without worrying that this may damage your body. Obviously, you will need to wear elbow protectors to practise elbow strikes with your sparring partner.
To deliver an elbow combo you will have to close the distance to your opponent. Elbows are good in combinations when you set them up with different strikes. You can connect with kicks and punches and pass the guard and get in range.
Try out and practise these combos:
left jab+right leg kick+left knee+right elbow
left hook+right uppercut+left elbow
My personal favorite is this one: jab-cross-left knee to the body-right elbow to the face.
It's simple yet effective and very intuitive. It is also a good way to get into a clinch.
For example: 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.
Elbows 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 situation outside of the ring. It is good to toughen them up with banana bag before you try it in a street fight.
Experiment with it - Practise it - and finally Implement it. There is no better weapon in Muay Thai arsenal to target your opponent's face from a very short distance.