An economizer is a device that allows to shorten HVAC unit run time and thus reduce energy costs when the outdoor air temperature (more correctly, heat) is practical to use.
For example, imagine a situation (happens quite often in climates with sharp day/night contrast) when the outdoor air temperature is already low, and the indoor air is still hot. In this case, it would make perfect sense to pump outside air in and dump indoor air out, rather than run the air conditioner.
Reverse situation: imagine an old brick or stone house, which is still cold in the spring, when the outdoor air is already warm enough to be used.
Note: See messages posted on the News Channel under economizer tag for more information about how it is supported.
Basically, an HRV is a fan with a heat exchanger: for cooling, the hot outdoor air will pass its energy to the air leaving the house, for heating, the hot indoor air will pass its energy to cold outdoor air entering the house.
HRVs only move the heat associated with air temperature difference. See the ERV section for more on this.
Note: Thanks to Jerry Scharf for his contribution to HRV and complete ERV section.