When a Twitter user 'follows' another, that user's messages will show up on the homepage along with messages from other users being followed. The number of followers a user has is a means of measuring that user's popularity.
Because Twitter messages are limited to 140 characters, it is difficult to include a standard URL, which alone can be much longer than the maximum 140 characters. This has given rise to various URL shortening services, which provide short links which are redirected to the intended target. These services have gained popularity outside of Twitter as well.