As far as I know, there could be as many as four possible types of check if we want to be as specific as possible:
1) a simple "active" check. I am not sure if there is a separate term for this, so let me call it simple active. So, a simple active check arises when the piece that moves gives check by itself but does not open any lines of attack from any other friend piece.
2) a discovered check. It arises when recently moved piece does NOT give check by itself but opens exactly one line of attack. A rider-type piece (being either a queen, a rook or a bishop) was hidden behind the piece that just moved.
3) a common double check where we combine #1 and #2: we have a piece that moved. It gave a check by itself. Also, it opened exactly one line of attack of a distinct frienly piece.
4) a very rare type of double check when the just-moved piece did NOT give check by itself. However, it opened TWO straight lines of attack, each one by a different piece but of the same type! I am pretty sure this is only possible during en passant. Also, both checking pieces must be queens of same color. One line of attack on enemy king must be a file and another, a diagonal.
However, there is one quite important complication. I have recently realized that a check in chess could arise from castling.
I analyzed all list items again, and it seems that this last case does not fall under any of #1, #2, #3 or #4.
So I am confused. The better you know chess rules, the more you get confused.
If a check arose from castling, would it be a discovered check (see # 2), or is it a totally distinct type of check?
Essentially, I want to add that the rook in this case is a very specific chess piece type — a hopper. https://en.wikipedia.org/wiki/Fairy_chess_piece#Hoppers