You can break a pin without moving the pinned piece by changing the conditions that make the pin effective. Typical methods include blocking the line of attack with another piece, attacking or driving away the pinning piece, or forcing a trade that removes it entirely. You can also break the pin by repositioning your king or valuable piece so that the pinned piece is no longer tied down. Each of these strategies dissolves the threat while keeping your position safe and coordinated.
Example 1:
Our knight is pinned to the queen. However, we haven't castled which gives us a little bit of flexibility when it comes to dealing with the pin: 1.h3 Bh5 2.g4!
and Black's bishop is forced to retreat to g6. The pin has been successfully broken! Since the position has a closed character, our king is not in danger if it remains in the center, but it can also go to f1 to stay even safer.
Example 2:
We are up a piece, but White has pinned our knight to the queen. Unfortunately, moving it isn't an option. Can Black do anything else to save the knight? If you were optimistic and said "Yes!", your intuition was correct as Black can play 1...Qa4!
White can't take the knight in any way, but it's still not over - White makes another attempt to force Black's queen to go back to where it came from: 2.b3 - luckily for Black, he has another trick up his sleeve: 2...Qe4! offering to trade queens one more time!
This time around White has to accept the trade as moving the queen would result in immediate checkmate: Qxe2 is coming! 3.Qxe4 Nxe4
Black was able to break the pin and save his knight in the aftermath.