I had to replace the shut off valve under the bathroom sink, well the little metal piece that prevents the adapter of the shut off valve from sliding off the pipe is broken. I need to know what the name of this piece is please, and where can I get a new one? Thanks in advance!

That "shutoff valve" is called a angle stop. The "adapter of the shut off valve valve" is called a compression nut and the part you're searching for is called a ferrule, (see image). Ferrules are made of brass and can be purchased from any plumbing shop. Ferrules are the part that compresses against the compression nut and the angle stop valve body to form a water tight seal,
Regards, Tom