The backstory of Mega Man X... I'm going by memory, so this is totally going to be all kinds of wrong.
It's the far flung future of 20XX (what does that even mean?) when Dr. Cain (I think that's his name?) unearths a capsule that had been sealed away for ages, containing the last, greatest creation of the late Dr. Light, X. A robot that can think and feel and grow and learn. Dr. Cain studies X, and reverse engineers him to replicate the technology, creating the Reploids. At some point Zero is also uncovered, who is the last creation of Dr. Wiley, and.. there's a computer virus that comes with him, I think? But he's cured of it, but it spreads to all the other reploids, and they start going maverick, and so a team of other reploids is put together to become maverick hunters, lead by Sigma, which is a great idea because if you have a computer virus turning reploids evil, you want to send more reploids after them, right? So Sigma turns evil too, and so do all of his animal-themed lieutenants, but X and Zero are immune and have to save the day.
And then right around Mega Man X 5, everything starts to turn to shit and none of this really matters.