Tankless water heater

I wonder how a tankless water heater works. With our current water heater, there's an overflow tank thing that my husband has to empty every once in awhile otherwise hot water pours out of it. Would that be avoided with a tankless heater? I've tried to read about it but it's awfully confusing. We live near Philadelphia so we do have cold winters where water would freeze and it looks like these go outside. Does that mean that only people in warmer climates can use them? They definitely look convenient though. I want one!