Predictive Text Entry, (also know by the most popular type, T9,) allows you to enter text by pressing only one key per letter. As you enter a word, the phone will automatically compare all of the possible letter combinations against a built-in dictionary of words, and determine which word you intended to type. If it guesses incorrectly, you can scroll through other possible words without re-typing the word.
This makes it much easier to enter long text messages and e-mail. The number of keystrokes is typically cut in half when using predictive text entry, compared to traditional methods, which require you to press 2 once for A, twice for B, three times for C, and so on for each key on the keypad.
There are a handful of competing technologies, including AOL's T9, Motorola's iTAP, and ZiCorp's eZiText. T9 is the most popular. eZiText competes directly with T9, and iTAP is used by Motorola in some (but not all) of its own phones.
There are no compatibility issues - a phone with T9 can exchange messages with an iTAP phone with no problem, and vice versa.