Footprints does a great job starting people out on ear training. I have some small bits to add. What has been talked about so far is melodic interval training. In other words, it's about hearing one note followed by another. This is definately step one. Octaves, fifths, fourths, etc... as stated above, are the way to start melodically. The next step is to move on harmonically.

This is really the key to start unlocking chords. When you hear a chord you should be able to pick out a note inside it and sing it loud and strong. Once you've got the melodic interval figured out, you should be playing them harmonically and singing the lower note, followed by the upper note of the interval. Sometimes this isn't as easy as you'd think. It's not too hard to screw up a harmonic fifth by singing a fourth melodic interval instead (i.e. instead of singing C-G, you sing G-C), or vice versa. After that you've got to start looking at triads and trying to sing the third, or some other piece of it. And, of course you keep getting more and more complex chords hitting various parts of them and singing them quickly, and perfectly.

The real master of ear training is David Lucas Burge. He sells a perfect pitch and relative pitch training course. It's quite expensive but it's really good! Take a search for him on the web if you need to find him. is a good place to get some more information on him and what can be accomplished with a good ear.