What exactly are humidity and dew point and how are they calculated?

Answer 1

Humidity is a measure of moisture content in the air. It is measured as a percentage. Higher percentages mean higher moisture content in a unit of air. 100%, for example, would mean the air unit is totally saturated with water. It's something like filling a glass with water up to its rim so that it is about to overflow.

Dew point is the temperature at which that moisture condenses out of the air. Higher moisture content, humidity, usually results in a higher dew point because the air has so much moisture content the moisture takes less cooling to condense. Lowering the air temperature is like lowering the rim of the glass; eventually the water will overflow.

Seen or not, air contains water. Warmer air can contain more water than colder air. Thus, for example, as unseen evaporation rises, the water, it tends to condense out and become clouds at higher altitudes where it is cooler. When those clouds rise even further, where it's really cold, they tend to join up and form rain, sleet, snow, or hail, depending on other conditions, like turbulence.

Answer 2

Humidity is the amount of water vapor in air. You can see the exact humity by using a psychrometric chart. All you have to do is to have a dry and wet bulb thermometer and then project the coordinates in this chart.
Relative humidity is the ration of partial pressure of water vapor and dry air. It is said that air is already saturated at 100% relative humidity. This means that the air could no longer hold any water vapor that tends to cling with it and if the temperature drops condensation will start and this is the dew point.

RH=Pwater/Pair.