You can customize your character based on the following:
Appearance (to a limited extent at the beginning)
Stats
Class
Job
As you level up, get in-game items from quests and enemies, buy items from the auction (uses in-game money), and/or use the Kafra Shop (requires real money), you can customize your character even more. Some of the items are rather odd and unique, such as some items that tend towards certain *ahem* fetishes (i.e., sailor fuku, goth loli costumes, the cat and bunny costumes, mushroom costumes, etc. (Yeah, some of these are pretty bad...Gravity seems to know their client base way too well...) You can get equipment and costumes, which work separately. Equipment changes your stats while costumes typically just change your appearance. So, once you max out your level (currently 50), you can focus on getting the best equipment to get the desired costumes you want by doing raids and random hard dungeons (RHD). However, most costumes are expensive, going for the thousands of zeny (think 1 million minimum gold). So, yeah...