Space Heating:

1. There are air-conditioners (エアコン) used for cooling, and most of the recent ones can also operate in reverse, to heat the room.  Technically, these are known as "heat pumps" in English, since they can pump heat into or out of the room.  These are very efficient, but still can draw a lot of power.  You can find these in other countries, though the styles common in Japan are different.  (Window based types and central types are almost non-existent.  Usually there is a per-room set-up with an indoor and outdoor box.  This will require several holes in the wall and a place to set the out-door box).  Higher voltage (200v) models can be more efficient than 100v models.

2. There are Ceramic Heaters, Electric oil radiators, quartz heaters, electric fan heaters, etc., as you can find in other countries.    These run on electricity, eat up a lot of power, and obviously can't cool.  A lot of breakers in Japan are relatively low amperage and it's easy to blow a breaker in many houses by using a resistive heater and using a microwave, hair dryer, etc., at the same time.  With most electric plans, raising the amperage raises the base rate you pay every month for your electric plan.

3. In-floor heating - This is popular in newer higher-end apartments, as it simply makes the room (including your feet) feel warm without you realizing where the heat is coming from.  Basically these need to be built into the house from the start.  

4. Gas Fan Heaters (ガスファンヒータ) - These are powered by natural gas, and there are gas outlets in the wall of some rooms in many houses, so that you can attach these.  Gas fan heaters are efficient, powerful, and cheaper than electric heaters to operate.  The down-side is mainly that they do consume oxygen, so you either need to leave a window cracked open, or open the window for a few minutes every hour.  You will see many rooms that have a gas outlet also have a small sliding sub-window on the larger window, to allow just a little air flow - built with gas fan heaters in mind. The gas cords are designed so that if either end is pulled out the flow of gas stops.  The heaters will automatically stop if tipped over, etc.  (This is also common in some other types of heaters).  If you don't have gas outlets in a room, then you basically can't use a gas fan heater in that room.  Gas fan heaters typically require electricity for ignition and temperature control, but the draw only a trivial amount of power.  There are also gas fan heaters based on propane (LP) and butane, for areas without natural gas, however these are uncommon as kerosene fan heaters are more popular due to ease of fuel replacement. (Typically butane canisters are small, and propane is used for heating water and cooking - using it for space heating would mean that the large propane tanks would empty out more quickly, and the household would have to wait for the next delivery - whereas, running out of kerosine only means making a run to the gas station).  To be sure you have the right type of heater, consult the salesman, and be sure it is marked as "都市ガス用" (Municipal Gas Usage).  Models marketed by (for example) Tokyo gas, of course use the municipal gas.

5. Kerosene fan eaters.  (石油ファンヒータ、灯油ファンヒータ).  These actually come in mainly two types, the Fan type and non-Fan type.  The non-fan type is like you would see in the country-side in many countries, with a wick and a visible flame, and sometimes don't even require electricity.  We won't discuss those further since they are antique.  The fan type is a bit more sophisticated, requires electricity, and looks like a gas fan heater.  Newer kerosine fan heaters are clean burning and efficient, however as with gas fan heaters, they will use up the oxygen in the room eventually, and so the same rules apply.  Kerosine fan heaters are more of a pain because they burn liquid kerosine, which is stored in an internal metal canister.  When the canister becomes empty, it must be refilled.  Families often store the kerosine tanks outside for safety (and smell) reasons, which means a trip out in the cold.  Note that the colloquial name for these heaters uses the word "石油", which means just "oil", but you must be sure to fill them with "灯油" (kerosine), and not gasoline, etc.  

Water Heating:

This is an area that deserves special attention.  90% of households use gas for heating water.  Since energy is expensive in Japan and water heating often accounts for a large percentage of the household energy budget, there are all sorts of tricks used to wring the last bit of efficiency from the gas used.  The first is simple and used in 95% of all households.  There are one or more control boxes for the water heater, and the heater can be turned on or off from any of the boxes.  Before using the hot water, you must turn on the heater and wait about 30 seconds for the water to heat up.  When done, you can turn off the heater to save gas.  You can also adjust the temperature of the hot water from this box.  

There is usually a control panel near the kitchen sink and another one near the bathtub.  The one near the bathtub can often be used to have the tub automatically fill itself to the desired water level and temperature.  The water in the bathtub can also be kept warm indefinitely (or re-heated later) so that the water can be re-used.

The cheapest water heaters are just small boxes where the water passes through and gas is burned to heat the water up as it flows.  Fancier models are more expensive, but also more efficient, so that they save money in the long run.  The most extreme example is the Enefarm system marketed by Tokyo Gas.  This system is a rather large box (think small refrigerator), which generates electricity from the gas using a fuel cell.  The heat produced by the system is used to heat water that can be used for baths, etc.  This way, the same gas generates heat and electricity simultaneously, and the electricity generates costs less than the electricity bought from the power company.