Humidity is a measure of evaporated water in the air. The higher the humidity in the home, the more water vapor there is in the air, and the warmer you will feel. Our bodies cool by sweating and allowing the sweat to evaporate. Evaporation occurs most easily in dry air and becomes increasingly slowed by increasing humidity. As such, in humid air, our bodies are less able to quickly cool, and we consequently feel warmer.
In their most basic forms, central air conditioners cool the air in the home using coils that contain cold gas, such as freon, and a fan to pass indoor air over those coils. The air is cooled by the coils, and water vapor condensates on the coils, just as it would on the outside of a cold water bottle in summer. The water then drips down into a drip pan, which drains the water outside the home, usually via a small PVC outlet pipe on the side of your house or in your gutter.
Window air conditioners, or wall-units, operate in much the same way central air conditioners do. They pass room temperature air over coils filled with cold gas, which causes condensation to accumulate on the coils, reducing indoor humidity. The condensate drips from the coils down in to a pan, which drains outside the home. This is the source of the water that drips from your window or wall unit and why some air conditioners will rust when their drains back up and water remains in the pan.
While your air conditioner does dehumidify your house, it is something of a side effect of the method in which it cools. If you have a serious humidity problem, as may be the case in a basement or root cellar, you may want to supplement your air conditioning with a standalone dehumidifier. Most modern air conditioners continue to run as long as the indoor air temperature remains above the thermostat setting or the indoor humidity remains above a certain level. By supplementing your cooling system with a standalone dehumidifier, you can reduce power usage by enabling your air conditioner to turn off sooner.