The definition of humidity can be very complex. There are also two entirely different concepts, absolute humidity and relative humidity. The measurements given in the weather forecasts are relative humidity.
The Relative Humidity expresses how much moisture is in the air, as a percentage of the total moisture the air could contain at the current temperature. In other words, if the air has all the water vapor that it can contain at a given temperature, the relative humidity is 100%.
However, the warmer the air, the more water vapor it can contain, and the cooler the air, the less vapor it can contain. (This is the reason moisture will condense on a cold glass window as the air nearby cools, it reaches the point where it can not contain all the water which it held at a warmer temperature, and some moisture has to come out of the air). Therefore, for a given amount of water vapor in the air, the relative humidity will change inversely with temperature: as the temperature drops, the air becomes relatively more saturated, and the relative humidity goes up. This means that the relative humidity can vary, as the example you gave, just because the temperature has changed, without there necessarily being any difference in the actual amount of water in the air.
Pressure also plays a small part in all this, so its not easy to make much sense of the humidity readings alone. In this respect, a more important measurement is the Dew Point. This is closely related to the Relative Humidity and is usually also given in weather reports. Dew point is the temperature the air would need to be cooled to to achieve 100% saturation and for water to condense from the air.
In general, when the relative humidity is high (the air is nearly saturated with moisture) the dew point will be close to the air temperature (at 100% Relative Humidity, the dew point is equal to the air temperature). When the relative humidity is low, the dew point is much lower than the air temperature.
If the amount of water in the air remains constant, the dew point will be steady, even as the temperature changes. Conversely, if the dew point is rising, more moisture is being absorbed by the air. So the best understanding of relative humidity is to compare the difference between the Dew Point and the actual temperature.
The Dew Point is a much better measurement than relative humidity and a good measurement of how “sweaty” you are going to feel. Humans are sensitive to humid air because the human body uses evaporative cooling as the primary mechanism to regulate temperature. Under humid conditions, the rate at which perspiration evaporates on the skin is lower than it would be under dry conditions. Because humans perceive the rate of heat transfer from the body rather than temperature itself, we feel warmer when the relative humidity is high than when it is low.
For example, if the air temperature is 75 °F and the relative humidity is zero percent, then the air temperature feels like 69 °F. Whereas, if the relative humidity is 100 percent at the same air temperature, then it feels like 80 °F. In other words, if the air is 75 °F and contains saturated water vapor, then the human body cools itself at the same rate as it would if it were 27 °C and dry. (See http://en.wikipedia.org/wiki/Relative_humidity#Comfort
The Dew Point and Relative Humidity are also responsible for many weather conditions, such as rain and fog. Clouds are formed because the air temperature decreases with height until it reaches saturation when the temperature is reduced to the Dew Point. On a hot Summer morning, the temperature rises at ground level and progressively heats the air above until the hot air reaches the cloud base, this is when summer storms occur.
The Dew Point is also, of course, the temperature at which dew or fog forms when the air cools suddenly. When the temperature of a moist surface layer of air drops, the air becomes saturated so the air contains all the water vapour it possibly can. Fog forms if it cools any further.
The attached chart (also from Wikipedia) shows the relationship between the temperature and the saturation point of water vapour.