While it's possible to use LiveJournal with just a web browser, it's convenient to download a small program that lets you work with your LiveJournal directly. For more information on what a client is, check out the FAQ.
First, pick your platform:
LiveJournal clients are available for a variety of platforms. If no version is available for your computer, or you don't want to download anything, you can still use the Web update page.
Clients written for the Blogger and Atom APIs should be able to work with LiveJournal, provided you can change which server they post to.
For information on how to develop your own client, check out the developer section.