What should I do

Take a piece of paper and draw a big T on it. That is, make two columns. Label one of them Reasons to Stay Together. Label the other one Reasons to Break Up.

Start filling in your lists. Pay attention to how quickly you think of reasons for either side, and how many you think of. You'll probably find your path is pretty clear.

If it isn't, consider counseling, either alone or with him. Can couples recover from infidelity? Yes, some can. But it's a ton of work. This is why I am suggesting the two-column exercise first. If it's not worth it for you to hang around, then don't bother.

Did he have a real physical relationship with this woman - or an internet affair?

Actual physical

Well this is such a critical situation for everyone in love because Love is too strong, faithful and full of emotion. Every one love to there partner but this kind of mistakes & attraction happen with love partner.

You should forgive him & should give another change but if again he does wrong in future than there is no changes to forgive me. Just give him one and last change & Be Happy too !

Trust has to be earned. One way is over time. And he can begin right now by staying off FB.

Only you can decide if his other characteristics are worth all this heart-ache.
The fact that he treats you so nice, yet cheats on the side indicates he's sneaky.

(I can't believe how much time that took for him to go from FB postings to a full blown physical affair. This guy has too much computer time on his hands.)