The name Tai Chi Tai Ji in Chinese / Japanese...

太極拳 is the famous Taoist meditation and martial art exercise. The direct translation of these characters would be something like “grand ultimate fist,” but that does not quite hit the mark for what this title really means.

An early-morning walk through any city in China near a park or an open area will yield a view of Chinese people practicing this ancient technique.

A typical scene is an old man of no less than 80 years on this earth, with a wispy white beard and perhaps a sword in one hand. He makes slow moves that are impossibly smooth. He is steady-footed and always in balance. For him, time is meaningless and proper form, and technique is far more important than speed.

For the younger generation, faster moves may look impressive and seem smooth to the casual observer. But more discipline and mental strength are needed to create perfectly smooth moves in virtual slow motion.

Note: There are two ways to Romanize these Chinese characters, as seen in the title above. The pronunciation and actual characters are the same in Chinese. If you really used English sounds/words to pronounce this, it would be something like “tie jee chew-on” (make the “chew-on” one flowing syllable).

太極拳道 is the common Tai Chi Chuan title with “Dao” (the Way) added to the end.

If you're not sure, I suggest shorter titles such as “Tai Chi Chuan,” or just “Tai Chi.”

太極 is the shortened title for Tai Chi Chuan or Tai Ji Quan that is sometimes used in Western countries.

Basically just removing the last character which means fist. I don't recommend this two-character selection because it's not really a word without the third character in Japanese and Chinese.

Peng, Lu, Ji, An, Cai, Lie, Zhou, and Kao are the eight fundamentals or forces of Tai Chi Chuan or Taiqiquan.

棚 (Peng) refers to the outward (or upward) expansion of energy.
履 (Lu) is often referred to as “rollback.” Lu is the ability to absorb, yield/deflect incoming force.
擠 (Ji) is often thought of as a “forward press.” However, it is also best described as a “squeezing out of space.”
按 (An) is a downward movement of energy, best translated as “(relaxed) sinking.”
採 (Cai or Tsai) translated as “downward pluck.” Cai is a combination of Lu and An.
列 (Lie or Lieh) is “Split” and is a combination of Peng and Ji.
肘 (Zhou) Elbowing.
靠 (Kao) Shouldering (for when the arms are bound/distance is too close to punch).

Source: https://combativecorner.wordpress.com/2015/12/03/the-8-energies-and-5-movements-of-taijiquan/

太极拳法 literally translates as “Tai Chi Fist Law” though 拳法 is also known in Japanese as “Kempo” which is sometimes read as “boxing” depending on context.