Accessing World of Warcraft data from the web
Sites like Wowhead and WoWHearoes use client run addons from players which collect data. The data is then posted to their website. There is no way to access WoW's database. Your best bet is to hit the armory and extract the XML returned from your searches. The armory is just an xml transform on xml data returned.
Blizzard has recently (8/15/2011) published draft documentation for their RESTful APIs at the following location:
http://blizzard.github.com/api-wow-docs/
The APIs cover information about characters, items, auctions, guilds, PVP, etc.
Requests to the API are currently throttled to 3,000 per day for anonymous usage, but there is a process for registering applications that have a legitimate need for more access.
Update (January 2019): The new Blizzard Battle.net Developer Portal is here:
https://develop.battle.net/
Request throttling limits and authentication rules have changed.
Characters can be mined from the armory, the pages are xml. Items are mined from the local installation game files, that's how wowhead does it at least.