A drudge is someone who works very hard, especially at a job that is boring. To drudge is to work hard.
One thing just about everyone has in common is complaining about work. Drudges have a little more to complain about than others: a drudge is someone whose job is difficult but isn't very interesting. Factories and coal mines are full of drudges: diligent folks doing work that isn't super-interesting. Such work is often called drudgery. Also, when you're working hard at anything, you can call it drudging. We all drudge when we have something to do that we would rather avoid.