Call them tankless, on demand or instantaneous, they provide hot water when it's needed, where it's needed - without holding any in storage. Usually electric water heaters or gas fired, tankless water heaters have the potential for big money savings.
Benefits of ditching the tank
Most of those savings come from eliminating standby losses, the energy that's wasted when heated water sits unused in a tank. And since there's no storage, you won't have to run the water waiting for it to get hot. A further benefit to eliminating the storage tank is avoiding leaks. Most storage tanks will eventually leak and can cost a lot to fix, whereas leaks rarely occur in tankless water heaters.
Some tankless water heaters are either completely ON or completely OFF. As flow rates change, they continue to put out the same heat. At the least, this can cause uncomfortable temperature fluctuations and at the worst, scalding. Modulated units base their heat output on the volume of water flowing through. Less water means less heating applied, keeping temperatures even.
Intermittent ignition device
The IID is like a spark ignition device on an oven. It ignites the pilot light when needed. An instantaneous water heater that has a continuously ignited pilot light can waste energy, so look for an IID to save energy.
Will it cover your family's needs?
Residential-sized (gas) tankless units can be great for homes with one or two people. In big homes with a lot of hot water needs, they might not provide enough.