I think if you want to do backstory well (whether in the form of a flashback or not), you have to put it in because you think the reader wants to know it, not because you think the reader should know it. You can't put it in and be like 'well, it might be a bit boring for people right now, but it'll be important later on'. I mean.. it
might be important later on. But then, you have to figure out some way of making it so that the reader wants to know it, too, right then.
The only exception I can think of is if the backstory piece is very, very short, like only a sentence long. Then the reader doesn't have any time to get impatient with you. * grins *
In The Hunt For Red October, there is a scene that reveals the identity of an espionage informer. It didn't actually say that he was the espionage informer, and a couple of my family missed it when they read the book, but I was watching for it. The thing was, you knew certain facts about the informer, you just didn't know much, or know his name. And then you came to one of the Russian scenes, and there was this guy, and the narrative starts talking about him, and telling some of his backstory, and some of his habits, and I guess if it had been talking about someone else, it would have been a little boring... But I recognized who it was, and as the narrative went along with the backstory, I only got more excited, because I got more and more sure that it was him (I had had several false alarms before, wondering if some random character was him).
Anyway, that was a good place to put in backstory, because I really wanted to read it. It revealed the identity of the informer and increased the tension of the scene and the next few scenes following--and, I had been wanting to know about that guy and his backstory for half the book.
I don't think I have anything to say about backstory relevancy, because I think Karthmin said everything pretty well.

He is right about how the past of a character is the only way to see what the now means properly. It makes things fuller and bigger. The entire first piece of The Count of Monte Christo is really backstory for the rest of the book. It wouldn't work right without it. Everything goes back to that very first piece, and the people in it, and what they did.
Karthmin wrote:
If you're talking character motivation (why he does what he does), then definitely keep it. Especially if it gives the character more history in the process.
I agree with this. One thing though: the longer the piece of backstory is, the later you might want to put it in the book, to give people time to be curious and get thinking about it.