Paris, 1750. Légal has a knight on f3, his opponent’s bishop on g4 is pinning it to the white queen on d1, and he moves the knight anyway.
8 
7 
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a c 
d 
e h 1 A queen for two pawns, and mate on move seven. Which is the whole answer in one game: whether you may move a pinned piece depends entirely on what it is pinned to, and whether you should depends on the arithmetic.
Pinned to the king: the move does not exist
No move may leave your own king in check, so a piece pinned against the king has no legal square off the pinning line. This is not a matter of judgement or a bad idea you talk yourself out of. The move is not in the list. A board refuses it, an app greys it out, and over the board it is an illegal move with a penalty attached.
7 b Count Black’s options in that diagram before you agree with me. King moves and the h-pawn, and that is the entire list. The knight is the worst piece in the set to have pinned, because a knight can never stay on a line, so pinning one against the king takes away every square it has at once. That is why a pinned knight attacked by a pawn is usually just lost, and why the pin is the second motif I teach, straight after the fork.
Along the line is still allowed
An absolutely pinned piece is confined, not frozen, and beginners give up moves they are entitled to. A rook pinned down the d-file may slide anywhere on the d-file, including onto the piece pinning it. A bishop pinned along a1-h8 can travel that diagonal. A pawn pinned straight down its own file can still push forward, since forward is along the pin. A knight always loses everything, and so does any piece pinned across a line it cannot travel: a bishop pinned down a file has no move at all.
Relative pins are an opinion, not a law
Pinned to a queen, a rook or anything short of the king, the pin is advice. You may move the piece whenever you want. Usually you do not, because what stands behind is worth more than what you would gain, and that judgement is just piece values plus a look at the resulting position. But the piece can leave with check, or take something bigger, or, as Légal showed, walk into a mate that makes the queen irrelevant. The full story of that game is on Légal’s mate.
Touch-move and the piece that cannot move
Art. 4.3 makes you move a piece you deliberately touch, but only if it has a legal move. Touch a knight that is pinned against your king and it has none, so you are free to play anything else. Touch a relatively pinned bishop and you are moving that bishop, whatever it costs you, which is a good reason to say j’adoube out loud before you straighten a piece that is sitting crooked in its square.