This is an expression used mostly in informal contexts. When someone in a difficult situation decides to cut and run, what he does is to quickly run away from his problems. Instead of dealing with them in a responsible way, he attempts to escape from them by leaving suddenly. This is something that all of us do some time or the other.
Here are a few examples. *When the pressure began to mount, many in the management decided to cut and run. *Raju’s cowardly decision to cut and run was criticised by everyone.
The expression comes from the world of sailing. In the old days, when a ship that was at anchor came under attack, what the sailors did in order to make a quick getaway was to cut the rope to which the anchor was tied. Instead of going through the lengthy process of raising the anchor and then setting sail, they merely “cut” the rope and allowed the ship to “run” before the wind. The crew lost the anchor by doing this, but this gave them a chance to escape. Nowadays of course, it is no longer possible to “cut and run” because anchors are attached to chains and not ropes!