A borrowed neutral is where something (nearly always a light) has its live connection on one circuit and its neutral on another:

This is most commonly encountered with 2-way switching of a landing light. The live originates on the downstairs lighting circuit and goes through the downstairs and upstairs switches to the landing light, where of course a neutral is also needed, and the easiest place to get one is from the upstairs lighting circuit. It was done like this for years, with never a problem because people only had 1 lighting circuit. The problem arises when the lighting is split into two - Upstairs & Downstairs.

The reason it's a problem is shown below. Consider the two circuits as before, and you want to break into Circuit 2 at point X to install a new light. So you switch off the MCB, or pull the fuse, verify that the circuit is dead, and cut the cable. At that point some or all of the neutral cable in Circuit 2 becomes live via the path through the light from Circuit 1.

I submit that this is a Bad Thing™