Tai chi chuan, T'ai chi ch'üan or tàijíquàn?
Chinese characters generally represent the meanings of words rather than their sound.
(Our symbols for numbers do the same thing.) We use our alphabet to represent the
sounds of the words -
Chinese uses sounds that we don't, and distinguishes sounds that we think are the same. 'Chinese' is eight different associated languages or dialects, which do, in fact, all sound different.
Chinese is a tonal language: whether the pitch of your voice stays level, rises or falls, and so on, makes a difference in meaning to an otherwise identical sound. Mandarin has four tones, Cantonese six, for example.
The apostrophes indicate a difference in pronunciation and the two dots over the u also make a difference.
Westerners have tended to ignore these extra marks when writing Chinese down, resulting in the simplified tai chi chuan, which is still widely used today. This is inaccurate, because ch and ch' are slightly different,* so we tend to confuse the chi in Tai Chi with what should be ch'i in Chi Kung. (See the Glossary.)
Pinyin versions are perhaps more accurate but at the same time they are less easy on the Western eye.
T'ai chi ch'üan (WG) is taijiquan in Pinyin.
Pinyin uses accents over the vowels to represent tones: tàijíquàn, and you are much
more likely to see this, as well as Zhèng Mànq+ng for Cheng Man-
On this site we've gone for simplified Wade-
* The apostrophe shows the sound is aspirated -
If you're interested in how Chinese is represented using the Latin alphabet, then the articles at Wikipedia are a good starting point.