A-1 has the ball for a throw in and B-1 fouls A-2. What kind of foul is called Intentional foul or personal foul?
Confused on situation.

Ball is live. Personal.

Either, depends on the nature of the foul. An interesting
question might be is this a technical or not. This is not
a dead ball foul so it's not a technical, but it could
still be either common, intentional or flagrant.

You're confuysed because you imply (or I infer) that Intentional and Personal are different categories of fouls.

They're not.

There are two basic categories -- Personal (live ball contact) and Technical (live ball non-contact and all dead ball).

Within these categories are various types -- common, player control, intentional, flagrant, double, multiple, false double, false multiple, ... Not all types fit in both categories (there's no such thing as a player control technical foul, for example). You also need to add a type of "none"

Each foul is one type, one category -- common personal, or flagrant technical, or "none" technical (we just call it "technical"). You can't have two types -- no intentional flagrant technical, for example. You can't have two categories -- intentional personal technical, for example.

In your play, the ball was live, contact was made, so it's a personal foul. Your question really should be, Is it common or intentional?

You (probably) have no difficulty deciding this during "playing action" at the end of the game. Use the same criteria during the throw-in.

He may also be confused by the rules about what happens if the foul is against A-1, while he is still out of bounds with the ball.

Chill--those rules apply ONLY to the fouls on the player throwing the ball in. All the other players are judged by the regular rules for contact.