A lot of perennial plants throw out shoots from the base of the plant as the season progresses. Left to grow, these obviously makes the plant more bushy for next year.
But as the name suggests, you can remove these shoots by cutting them as close to the base of the plant as possible, trim with a sharp knife, and plant into usually a gritty compost in a greenhouse when they will form roots eventually and give you a new plant.
That's the logic behind it. I'm sure you will get a few more replies as to what sort of plants are suitable and when to take them.

I use basal cuttings for propagating small shrubs, such as daphnes, though they can also be used on herbacious plants. In spring/early summer I take new shoots, about a couple of inches long and gently 'tear' them from the plant so that they have a heel or base, hence basal. The thin end of the cutting is trimed and then inserted in a pot of sharp sand and placed in an unheated propagator in the greenhouse. I don't find that rooting powder or gel does any good at all so don't use it. The sand is kept moist but not wet and I put a small pot of water in the propagator to keep the atmosphere humid.