1. To become thinner or narrower, especially at a gradual degree. The giant wall went on for miles, eventually tapering off into a small strip of stones deep in the countryside. By tapering off to such a fine point, the needle is able to puncture the skin without causing much pain.

2. To cause something to become thinner or narrower, as by grinding or whittling its surface. In this usage, a noun or pronoun can be used between "taper" and "off." We'll have to taper off one end of the replacement rod to get it to fit into the machinery.

3. To weaken, lessen, or diminish gradually. The storm is expected to taper off later in the afternoon. You're going to feel very groggy at first, but the effects should taper off after a couple of days.

4. To gradually reduce the amount of something one ingests or some activity in which one partakes with the aim of stopping entirely. In this usage, a noun or pronoun can be used between "taper" and "off." I was on a lot of medication following the surgery, but I'm finally starting to taper that off. Going cold turkey didn't work for me, so I'm going to try tapering off my smoking instead.