Heaters
If you are keeping tropical fish, you will need a heater. A heater insures that a tank doesn't get too cool, and that the temperature stays steady during the course of the day, even when the room cools off (e.g., at night). For many tropical fish, a temperature of 78F is ideal.
There are two main heater types. Submersible heaters stay completely below the water. A second, more traditional style, has a partially submerged glass tube (which contains the heating coils), but leaves the controls above the water. Submersible heaters are the better design, as they can be placed horizontally along the tank's bottom. This helps keep tank temperature uniform (heat rises), and prevents the heater from becoming exposed while doing partial water changes. With the traditional design, one must remember to unplug the heater before doing water changes; if the heater is accidentally left on while the coil is above the water, the tube gets hot and may crack when you fill the tank back up with water.
If your room is never more than 8-10F degrees cooler than your target tank temperature, a heater of roughly 2.5 Watts per gallon will suffice. If the differential is higher, up to 5 Watts (or more) per gallon may be necessary. Remember, the heater needs to keep the tank at its target temperature, even when the room is at its coldest point; the tank's temperature should not fluctuate.
Heaters (especially cheap ones) will fail. Most often the contact that actually turns the heater on and off gets permanently stuck, either in the on or off position. In the former case, your tank can get VERY hot, especially if the heater is larger than your tank actually requires. To minimize potential problems, avoid heaters larger than the optimal size for your tank. To prevent winter disasters, use two smaller heaters in parallel rather than one large one. That way if one fails, the consequences won't be as disastrous.
Thermometers
You will need a thermometer to verify that your tank stays at its proper temperature. Two types are commonly available. The traditional bulb thermometer works the same way as the ones you can buy for your house. They either hang from the top edge of your tank, or float along the surface. The second common design is a flat model that sticks to the outside of the glass. In this design, liquid crystals activate at a specific temperature, either highlighting the numerical temperature or a bar that slides along a scale.
Aquarium thermometers can be rather unreliable (check out the ones on display at a fish store --- they should all register the same temperature, but frequently don't). Thus, thermometers are good for verifying that your temperature is not too far off, but may be off by several degrees in some cases. When buying a thermometer, look at all the thermometers and pick one that has an ``average'' temperature, rather than one of the extremes.