Technically: Twitter is a service that forwards SMS text messages to subscribers who register to receive them.
Business: If sent and received by phone, sender pays for one SMS, per tweet, each subscriber pays for one to recieve it. Since many U.S. mobile users are already paying for unlimited SMS, number of tweets doesn't matter to them. Twitter account holders have the option to initiate and view tweets on Twitter's web site.
Social function: quite varied. Twitter is used to deliver political messages (the White House staff tweets as the President, most campaigns now tweet, but it is typically only their political followers and press that follow the tweets). Celebrities tweet running accounts of daily activities and have huge followings. Small social groups tweet to stay in touch, coordinate, commiserate. Emergency services tweet to coordinate activities and report situations, as Twitter turns around the SMS almost immediately and it goes out as quickly as the mobile phone provider can handle the message (SMS routes on the network that constantly tracks what mobile devices are in a cell and communicating).