Only being able to move and not capture while in check I was playing chess with my Dad, and he was shocked when I made a capture while check. He said that when he learned to play, that, when you are in check, you cannot capture with the king or any other piece to get out of check. This means that all you can do is move out of check. Is there a special rule or game type where this is true? We live in Hungary, so maybe they play different here.
Below is for regular chess. I don't know about any chess variant which would have a rule like you describe. 

If you are in check you have to deal with it somehow, i.e. after your move you should not be in check anymore. If there is no move which achieves this, it is check-mate.

There are at most three different ways to get out of check:


move the king out of check
capture the piece which attacks your king
put a piece in-between the piece which attacks your king and your king (this can only work if a bishop, rook or queen attacks your king)


A simple example:



White is in check by the rook on h7. There are three possible moves for white: Kg1 (see "1"), Bxh7 ("2"), Nh3 (see "3").

As a side note, you cannot castle out of check.