Many thermostats can control many of the most common types of residential HVAC (Heating Ventilation Air Conditioning) systems.
Below are some of the various types of systems you might have installed in your house.
Single Stage (most common)
"Single Stage" means that you have 1 source of heating (typically a gas/electric/oil furnace), and 1 source of cooling (typically an outdoor air conditioning unit).
"Multi-Stage" means that you may have more than 1 source of heating and/or cooling.
Typically there is a primary source of heating/cooling, and 1 or more backup sources.
Some air conditioners have multi-stage compressors to save energy when small amounts of cooling are needed.
Likewise some furnaces have multiple stages of heating.
A Heat Pump is basically an outdoor air conditioning unit that can both heat & cool by reversing the compressor.
Heat Pump systems may require additional stages of heating (and sometimes cooling) because when heating they sometimes require a little help when it's very cold outside.
Heat Pump systems require a thermostat that can not only control heating & cooling, but that can also switch the compressor from cooling to heating (called "changeover" or "reversing valve").
The "changeover" connection terminal is typically labled "C/O", "O", or "B".
A "Line Voltage" system operates at high voltages.
This means that instead of sending low-voltage control signals, a Line Voltage thermostat can control heating elements directly high-voltage (typically 220V).
The voltages in these systems can kill.
If you have a system like this, you should contact a professional for thermostat replacement.
In some geographic areas there is little or no cooling requirements.
A Heat Only system means that there is only 1 source of heating, and no source for cooling.
Millivolt (Heat Only)
A Millivolt system means that the control signal used to turn on/off the furnace is generated by a special thermocouple (device that produces electricity based on temperature) placed in a furnace's pilot light.
The very low-voltage signal is passed through a simple switch in the thermostat that uses the low-voltage signal to turn the furnace on or off.