I've trained in Muay Thai a bit and a few of my friends fight professionally here in Thailand. Some practice Muay Thai Boran which is the older artform as opposed to the modern sport version. There are way too many techniques to even begin to describe.
The principles are simple. Stay relaxed, conserve your energy, don't telegraph your strike, never show pain and keep your head verical. Most of the juicy stuff in Muay Thai can be traced back to Muay Thai Boran (Ancient Muay Thai) so I'll describe the basic drills from this art.
First let me say that in Muay Thai Boran there were a whole arsenal of throws entered into by catching kicks, grasping the wrists or from clinch fighting. Three of the four drills I've listed were aimed at throwing.
The fists were bound by a thick cotton twine, sometimes almost halfway to the elbow, but the palms were deliberately left open. The glass & glue were exceptions. Usually it was just the twine and grasping with the hands was prevalent.
There were 4 basic "sensitivity" drills involving the hands and each school had their own additional drills. They covered the striking, knee & elbow, and the standing grappling ranges. They are practiced offensively for the most part.
These four would be classified as Mae Mai (mother tricks). These and other techniques are the "basic package" that all Thai Boxers knew. Other drills and techniques unique to your teacher or your training camp are called Luuk Mai (offspring's tricks). These are too numerous and varied to be discussed here.
The first drill covers the striking range and can train kicks, punches and elbows. Both fighters hold up their fist and touch each other's. Right touching opp's left & vice versa. From here either would launch a strike and the other defend it. It's usually a one-off strike. It also helps train distance. Interestingly on old Muay Thai often the kick was thrown only after one of your hands had made contact in some way with the opponent. Dropping the lead hand during a round kick is a newer method adopted in the last 20 years or so to add more power to the round kick.
The second and only surviving drill is "neck wrestling". If you've ever played this with a professional Thai Muay Thai trainer you'll feel like your in a push hands session working on a specific skill. The idea is to grab the back of the head with either or both hands, placing the bone at the base of the skull (forgot the word for it) in the palm and pulling the head down as far as possible. When pulling with one or both hands the elbows are always pulled downward and inward. The opponent defends. Sometimes knees are included but mostly not as the drill is to get the head down. The object is to use a "sticky hands" type of movement to defend the opponent even reaching your head and enabling you to catch his. It gets pretty aggressive sometimes.
The third trains a skill that is not legal in the ring anymore. It lost popularity primarily because you get no points for this type of throw. It resembles wrestling's pummeling drills. Each fighter would try to grasp the opponent with both hands low on the back around the kidney area and squeeze the •••• out of him. Simultaneously he would drive the point of the chin into the opponent's chest just below the collar-bone pushing the opp's upper body back while pulling the lower body in thereby throwing him. In old Muay Thai Boran he would be allowed to finish him on the ground with a knee or elbow.
The final drill was to develop the skill of catching the opponents wrist (offensively or defensively) or kick (defensively). Grasping the wrist was a very useful skill in Muay Thai Boran to set up strikes and throws. As mentioned earlier most strikes were made after contact has been made with the opponent and often after catching the wrist. It's trained pretty much as you would imagine. One of the defenses to the wrist catch, strangely enough, is a rolling kind of elbow strike. When the kick is caught the defense is either to turn the knee out and pull the foot away or pull the opponent in and strike him. Although just catching it in the drill was enough.
Most people who've seen Muay Thai think the training must be brutal. It's not. The Thais have a different sense of time. Most Thai boxers know they'll be spending all of their years in training. They don't rush and each boxer has his own pace. Very much like in a Boxing gym. Another thing that surprises most westerners is their playful attitude and seemingly lack of machismo. The training is definitely hard but there's a surprising lack of tough-guy attitudes in a Thai Kai Muay (training camp). These guys know firsthand that there's always someone better. No one retires undefeated here.
If you keep in mind the drills I've mentioned while watching Muay Thai you'll see the remnants of these training methods - not backing up, one-off strikes, frequent touching of the gloves, neck-wrestling, catching the kick, etc.
Muay Thai Boran is a wonderful dying martial art. The average family income in Thailand this minute is 3,500 baht per month. That's around $80. Most Thai Boxers are poor and fight for money. There's no money to be made in Ancient Muay Thai. Unfortunately it's dying. There have been too many generations of Muay Thai fighters using the sport rules we see today for a resurgence and only a handful of teachers who know Muay Thai Boran left.