see
go out and its various conjugations. when you dump someone, you
cease to go out with them. this also is known as
breaking up, although breaking up tends to have more in the way of
agreement over the matter.
generally, you dump someone when they do something really bad, such as cheating on you. not that dumping the person should be the default response.
when you dump someone, chances are that he/she will go through hell for a few weeks. if you break up, both of you will go through hell for a while; the duration probably varies more (i wouldn't know, fortunately).
just say no to dumping people. break up instead. if you break up, there's the minute chance that you won't hate each other forever.