I mean beyond what JKR said it was, like the author says he doesn't age or something and that he can go to other worlds/dimensions/etc.

How do they have Harry leave his world and enter a new one? Through the veil?

 

Not necessary. I remember the story (HP / LotR crossover) when Harry used the Three Hallows to power up the dimensional traveling spell.

 

Was it an actual spell, or did they just describe the effects?

 

A lot of the hp/avengers crossovers fit THAT, so does the hp/do crossovers stick of death, master of death, ice age and the one by water mage whose name I can't think of right now