Basically, before the Sack of King's Landing, Varys got some peasant boy from some mill somewhere and gave Aegon to Jon Connington, a friend of Rhaegar's and an exile lord to keep safe until he was old enough to retake the Iron Throne (Aegon was still a baby then).

Connington then smuggled Aegon around for about a decade or so, training him to be kingly, teaching him about "the mysteries of the Faith", fighting, history, etc.

After Varys helped Tyrion escape, he took him to Illyrio's manse, and then Illyrio brought him to this poleboat where Connington and Aegon were hiding. They were disguised as "Griff and Young Griff", because the sigil of Connington's House used to be a griffin (so subtle right?). They were on their way to Meereen, so that they could marry Aegon to Dany and then go to Westeros together.

Tyrion eventually figures out who they are, and manages to convince Aegon to sail west instead of east with this epic speech. So now I'm the books, Aegon has already landed in Westeros, and is currently about to attack Storm's End, while Varys is helping in King's Landing by killing people that are actually decent rulers and causing chaos in the capital.

And that's Aegon in a nutshell.