1. To crush or pulverize something. And then this machine will grind away at all of the cans.

2. To frequently nag or disparage someone. Quit grinding away at me, will you? I'm doing the best I can!

3. To work hard on something; to dedicate oneself to some effort or task. I've been grinding away at the budget all morning, but I still can't figure out where that discrepancy came from.