Tai Ji is an expression used by various Chinese schools of cosmology to express a powerful universal force. What exactly the Tai Ji is depends on what particular time period and school of thought it is interpreted in. There is also, in addition, an entire school of martial arts called "Tai Ji Chuan", because it was used as a form of moving meditation undertaken by Daoist monks while contemplating the Tai Ji.
The simplest explanation of what the Tai Ji is is that it is the highest union of opposites, being the state of being that underlies hotness and coldness, light and dark, and the other pairs of opposites loosely described by the concepts of Yin and yang.
The literal grammatical meaning of Tai Ji comes from the word "Tai", an adverb meaning "extreme" and "Ji", another adverb meaning extreme. In modern Chinese vernacular, both of these expressions would be very common in everyday speech to describe everything from the spiciness of your food to the hecticness of your work day. Tai is a regular adverb, occuring before the verb it descibrs. Ji is somewhat of an irregular adverb, occuring in the phrase ji le, placed after the verb it descibes. However, using both tai and ji le in a sentence would be ungrammatical, as it would be redundant, more or less the equivalent of saying something like "the day was extremely really hot". Thus, Tai Ji is a combination of two adverbs that shouldn't be used together, without a verb, which they theoretically need. Their meaning could best be understood, perhaps as "Extremely exceeding". This weird conjunction of two grammatical participles can perhaps be excused on the ground that the Tai Ji itself is a transcendent concept that can not be understood through normal means, and therefore has no use for common grammar.
Note: Tai Ji is often spelled in English "Tai Chi", as a holdover from the days of Wade-Giles Romanization. This, however, as well as being a somewhat obsolete way of spelling the term, also is confusing since it confuses the "chi" with the "ch'i" ("Qi" in pinyin) of Qi Gong and the like. The "Ji" of Tai Ji has nothing to do with the word qi, which means "vapor".