When you abrade something, you scrub it vigorously. To clean your dirty bathtub well, you'll have to abrade it with a scrub brush and powdered bleach.
When you scour something so vigorously that you remove dirt, grime, or even the surface of the thing you're scrubbing, you abrade it. You might need to abrade your fingers with a nail brush to get them clean after working on your oily car engine all afternoon. The word abrade is related to abrasion — a scrape on the skin — and both words come from the Latin root abradere, which means "to scrape off."