What if there's a touch move rule, and my knight or bishop is blocking my king, and I touch it. Is that a checkmate? If not, what is it called?
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No. The touch-move rule can't force you to make an illegal move, and it is not legal to put yourself in check.
Here's what the FIDE laws of chess say:
4.5 If none of the pieces touched in accordance with Article 4.3 or Article 4.4 can be moved or captured, the player may make any legal move
No, that's not a checkmate. The white bishop is currently pinned (in absolute pin because it protects the king from check by Black's rook), so there's no legal move can be made by that bishop.
In short, this situation has no consequences, and you still can touch the king to continue play since touch-move rule doesn't apply for pieces in absolute pin.
Note that the checkmate only occurs if:
the king is under threat (in check);
the checked king doesn't have any legal move to eliminate the threat (by capturing the attacking piece, moving to other square or interposing with other piece).
Similar issue: What happens if you touch a pinned piece?
I agree with those who responded that this is not a check mate, but rather the piece is absolutely pinned, and since it's not legal to move that piece, another move, if available, must be played.
I'd like to mention that, in tournament play, there are consequences to moving an absolutely pinned piece and then pressing the clock, since this is considered an illegal move:
See FIDE rules 7.5.1 and 7.5.5 (https://www.fide.com/fide/handbook.html?id=208&view=article)
For your specific game, if it's King + Bishop versus King + Rook and nothing else is on the board and no mistakes are made, then this is usually a theoretical draw.