I generally try to avoid layovers, but I suppose I'd try to get out of the airport and explore on a longer layover. Instead I'll tell you what I do to spend the time before flights (which could be done as a long layover).

For instance at DCA and ORD the train goes right to the airport. I've explored Georgetown on foot (while stashing my carryon at a hotel--but I could've kept it with me if neccesary) while waiting for the hour to become right to head to DCA. I suppose the Mall would've been good too. Maybe 20 mins on the DC Metro. Same with ORD, and I would check out the Jefferson Park area (a large Polish neighborhood) were I not already living here.

I can't wait for the light rail at SEA to get into service. I've never had reason to go to MSP, but the Mall of America is on the light rail there. Not much to shop for, but the people watching is phenomenal. LHR could be fun. AMS looks like it could be good for a long layover too.

I don't quite trust busses enough to pull that trick at airports that don't have rail service.

Otherwise, I hang tight and play the airport game. I sit and see if I can figure out who is headed out from home or going home as passengers board other flights.