I've concentrated on OpenOffice.org Writer perhaps because it is the real contender to Microsoft Office and the one that most people moving to Linux from the Microsoft Office world are likely to want to use. That's not to say that this is your only choice. For instance, the KDE suite comes with its own word processor, part of the KOffice suite. It is called KWord.
KWord is a frame-based word processing package. People who are used to working with desktop publishing packages such as FrameMaker will find this a familiar environment, just as those coming from Microsoft Word will experience somewhat of a learning curve. What KWord does is make it possible to create extremely precise documents where the layout of text and graphics must be accurate.
Another excellent word processor worth your consideration is Abiword.
You can probably find Abiword on your distribution CDs but you can always get the latest version on the Abisource Web site (http://www.abisource.com). What Abiword really has going for it is size and performance. This is a lightweight application that will perform well even on slower machines. It starts up fast and is excellent at what it does.
What KWord and Abiword have going against them (at least at the time of this writing) are compatibility issues with Microsoft Word documents. Both read the documents fairly well, but they do not export quite as well. As time goes on and development in import and export filters continues, this may not be an issue for long.