What does OP_CHECKSEQUENCEVERIFY (OP_CSV) do? Some people have suggested repurposing OP_NOP3 as a new opcode, OP_CHECKSEQUENCEVERIFY.

What would the new opcode do? Why is it useful? Can't the signer choose the sequence field when they sign?
OP_CHECKSEQUENCEVERIFY (a.k.a OP_RELATIVECHECKLOCKTIMEVERIFY) can do relative lock times (used for sidechains and lightning).
It basically checks, if an input's sequence number is smaller than the sequence threshold (1 &lt;&lt; 31 = 2147483648), and if so, it will compare the nLockTime against the sequence number of the input.
More info:

https://github.com/ElementsProject/elementsproject.github.io/tree/master#relative-lock-time
https://github.com/maaku/bitcoin/commit/fd9864c3a3c6f2942bc0775c6c725aa379f9d616
https://lightning.network/lightning-network-paper.pdf