Twitter the site is a micro-blogging social media web site. Think of it like group texting with a limit of 140 characters. What people do with twitter is varied. Some use it as a social bookmarking site, the post a short comment and a shortened URL, others use it as a way to update there community with a status of what they are doing.
Twitter search allows you to do a keyword search on all updates across the site.
Twitter the company is a privately funded start up with offices in the SoMA neighborhood of San Francisco, CA. Started as a side project in March of 2006, Twitter has grown into a real-time short messaging service that works over multiple networks and devices.