On a plant, an offshoot is a sprout or shoot that branches off. From this botanical meaning, offshoot has come to describe anything that starts or grows from something else.
You can start a whole new plant with a houseplant's offshoot, by putting it in water and waiting for it to grow roots. Another kind of offshoot is a local branch of a library, an offshoot of the central location, or an offshoot of a popular TV show, also called a "spin-off." Interestingly, the earliest definition of offshoot, in the seventeenth century, referred to family trees. It wasn't until 1814 that offshoot was used for actual plants.