I've recently been helping people learn how to sing harmony.  There are steps to take to get ready to sing harmony, such as learning to sing melody on pitch and learning the harmony part with a piano and some sheet music, but once you have followed those steps, you need to really listen to the other notes around you. 

Singing harmony by very definition means you are singing with other people, so do not attempt this alone.  Singing with other people is lots of fun and they can help you to know whether you are doing it right or not. 

When learning parts, you should learn all the parts, not just your own.  Know everyone else's part, too.  That way, you will know where you are in relation to them as you are singing.  When practicing other parts, take turns singing their part and singing your part as they sing their part.  When learning how to sing harmony, it is important to take the time necessary to really learn every part.  It's like learning a song 3-5 times instead of just once, so give yourself ample time to do it justice. 

To sing harmony well, you need to make sure you are not overpowering the other singers with your notes.  The melody is the most important note and should be heard above the other notes.  Since you're singing harmony, you need to blend with the other singers.   Sometimes singing in a circle if there are many performers will help you to hear all the other singers.  Although this is not usually done in a performance, it can really help in practice sessions.