When you call your mother, (you do call your mother, don't you? Well? Go call your mother right now! I'll wait) this is the thing that takes your call and either sends it to her telephone, if she lives nearby, or sends it along to a central office or tandem office, if she lives far away.
These switches are grouped into classes, depending on what they do. They are all capable of handling 4-wire connections (important for high-speed lines, such as a T1, T3, or DS3.) The class designations are also sometimes used to refer to central offices, with similar meanings.