Tai Chi Chuan 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.

五行太極拳 is a certain school or style of Tai Chi (Taiji).

The characters literally mean “Five Elements Tai Chi Fist.”

Notes:
In Taiwan, it would be Romanized as “Wu Hsing Tai Chi Chuan” - see the standard Mandarin method above in the gray box (used in mainland China and the official Romanization used by the Library of Congress).

The last three characters are sometimes translated as “Grand Ultimate Fist,” so the whole thing can be “Five Elements Grand Ultimate Fist” if you wish.

I have not confirmed the use of this title in Korean but if it is used, it's probably only by martial arts enthusiasts. The pronunciation is correct, as shown above for Korean.

太極詠春功夫 is the title Tai Chi Wing Chun Kung Fu.

Please note that it can be romanized in a variety of ways such as:
Tai Chi Ving Tsun Kung Fu
Tai Qi Yong Chun Gong Fu
Taai Gik Wing Ceon Gung Fu

Be sure this is the right title for what you are looking for before you order.

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.

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/

太極球 is the title “Tai Chi Ball.”

This may refer to any number of variations of exercise balls, some with yin-yang themes, others strictly for taichi/taiji or qigong exercises.