There is the original (and misleading) connection, as stated in Season 2, and then the true connection, as stated in Seasons 4/5.
Season 2:
Throughout this season we learn more and more about these special children. At first it seemed that all their parents had been burned by the Yellowed Eyed Demon (aka Azazel), but it soon becomes clear there are exceptions to this (for example Ava Wilson, whose mother was still alive). All we know about them is that they all have psychic abilities and the Azazel supposedly wants them to lead an army:
YED: I’m trying to help you. That’s why we’re talking. You’re the one
I’m rooting for.
SAM: What’s that supposed to mean?
YED: Welcome to
the Miss America pageant. Why do you think you’re here? This is a
competition. Only one of you crazy kids is gonna make it out of here
alive.
SAM: I thought we were supposed to be—
YED: Soldiers in a
coming war? That’s true. You are. But here’s the thing: I don’t need
soldiers. I need soldier. I just need the one.
SAM: Why?
YED: Well, I
couldn’t just come out and say that, could I, Sam? I had to let
everyone think they had a fighting chance. But what I need ... is a
leader.
SAM: To lead who?
YED: Oh, I’ve already got my army. Or, I
will soon, anyway.
Connection: All have psychic powers. One is meant to be the leader of Azazel's army.
Season 4/5 (Possible spoilers)
Once we reach Season 4/5, we learn the true purpose and connection between the Special Children. Azazel, in 1972, found the gates to Lucifer's prison and slaughtered a convent of nuns to open it. It wasn't enough however, and he was told to do two things: get Lilith, to destroy the 66 Seals, and find him a vessel.
That was the true purpose of the vessels - to be a vessel for Lucifer.
Azazel fed all of them demon blood when they were younger, which made them stronger (as Sam became stronger throughout Season 4 when drinking Ruby's blood. The mother's who were burned alive were those who walked in on the process. Those who didn't survived.
Azazel's test, at the end of Season 2, was to find the most worthy vessel. He would be the leader of the army, i.e. the vessel of Lucifer. Ultimately, despite his death in Season 2, Sam was left alive and thus the only true vessel for Lucifer (as all of Season 5 shows).
Connection: All were fed blood by Azazel. The strongest is meant to be the vessel of Lucifer.
