A remote-cutoff tube is one where a relatively large negative grid voltage is required to reduce the cathode current to near zero (cutoff). In practice, this is done with a grid winding where the spacing is not constant along its length. It is roughly equivalent to two tubes in parallel, where the tube with a tight spacing cuts off at a lower grid bias, but the other continues to conduct up to a higher negative bias voltage. This grid structure is commonly used in pentodes, triodes, and mixer tubes in RF applications for better control of gain with small input signals.
They are not used in ordinary audio circuits, since they are inherently more non-linear than the “sharp cutoff” variety. Typical examples: 6AU6 (sharp) / 6BA6 (remote), or the older 6SJ7 (sharp) / 6SK7 (remote). If you compare the plate characteristics graphed in the data sheets for these pairs, you will see the difference.
In a guitar amplifier, you might exploit the distortion of the remote-cutoff tube, which would have large even-order harmonics at high levels.