Perfect Pitch has some different Definitions:
There are different levels of perfect pitch and absolute pitch. They are all about being able to either identify or generate (i.e. sing) notes without a recent reference. People's ability range from being able to guess right some of the time, to instantly telling you which note is not being played when someone bangs 11 out of 12 notes as a cluster on the piano. I emphasized "recent" reference, because in many ways what most people call perfect pitch in a way is just relative pitch, except you are using relative pitches they you haven't heard even that day.
Generating Notes without Reference:
I'm going to focus on the ability to be able to sing a note when you first wake up. This is often a good starting point for people that "don't have perfect pitch". The reason is, and here is the kicker, many people do and they just don't know it.
The way you can discover this is by first thing in the morning, try think of a piece you were playing the day before. Think of what key that was in, sing the not of that key, and see if you are right. Yes, you are just guessing, but what many musicians who practice a lot discover is that they guess right quite often (not always, but more often than not).
If you look at my "ability to hear music before it is played" section in this answer, I talk about learning to sight sing by only playing the note of the key as a starting reference. If you "guess" by using your memory of a various pieces in different keys you know before you play the note, you might find you tend to guess right there as well.
From this, over time, many people start to develop some sense of what many people call "perfect pitch". I think it is part learning to trust your instinct, and part developing the ability. Like many things in learning music, this process will probably take years. I think the most important thing about trying to learn this ability is to have fun "guessing", and if it just never happens for you it is not really a big deal, many great musicians don't really have this ability.