If you do something complacently, you act in a self-satisfied, unconcerned manner. It's not a good idea to putter around complacently on the week your boss is giving you a performance review.
Complacently is the adverb form of complacent, which means "with pleasure or satisfaction." When we say someone is behaving complacently, they're doing something confidently, but perhaps without fear of potential dangers. So complacency isn't such a good thing. When Hansel and Gretel make a new friend in the witch who lives in a house made of candy, they are handling things rather complacently.