A tracker is a server that aids in communication between peers who download files off the torrent network. For a client to download files from the torrent network, its necessary that it connect to a tracker first. Without a tracker the client cannot initiate the download.
The tracker provides the client with information regarding the peers who has pieces of the file its trying to download. Once a client gets the information it will contact the peers directly and then download pieces from it. The client contacts the server periodically updating its status and getting information about new clients.
Trackers are of two types, public tracker and private tracker. Anyone can connect to a public tracker and get information regarding the files it tracks. But private trackers are password protected. Private torrents are used to maintain the quality of clients. Some private trackers are invitation only. Some have upload ratio that needs to be maintained by the client to be able to be a member.