In contemporary chess notation a single plus sign + after a move signals check while the hash symbol # shows checkmate.
A double plus sign ++ is rare in modern databases and when it appears it tells the reader that the move delivers double check, two pieces attack the enemy king at the same time.
Why the double plus still appears in books
Early English manuals used descriptive notation. In that system + meant check and ++ meant mate.
When algebraic notation became universal during the nineteen seventies FIDE adopted # for mate. Some publishers kept ++ for double check to recycle diagrams without altering commentary. This is why you may still encounter both signs in classic game collections.
How double check works on the board
Only three tactical patterns create double check.
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A discovered check in which the front piece also gives check after it moves.
Example: bishop b2 protects the long diagonal toward g7. Knight f6 steps to h7 visiting g5 along the way. Queen g4 is uncovered and also checks g7. The king receives two threats. -
Castling through a blocking piece. White castles long while a rook on d1 uncovers a check from the queen on d3 and the rook itself gives check from d1 at the same time.
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A rare promotion where the new queen plus the promoted pawn both attack the king immediately.
Because two lines of fire converge on the king there is only one legal reply. The king must move. Capture or interposition will always leave one line open.