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. www.eartraining.com is a good place to get some more information on
him and what can be accomplished with a good ear.