Please bear with me on this long explanation, as it will hopefully make sense.
The air is capable of holding only so much moisture (i.e. water vapor) for a given air temperature. Because this amount is dependent on temperature (and pressure too, but let's ignore that factor), the amount of moisture in the air is stated as percent Relative Humidity (RH), "relative" being in relation to the temperature. The key thing to understand is that, the higher the air temperature, the greater the amount of water vapor can be held in the air...and conversely, the lower the air temperature, the less water vapor can be held in the air. For example, say it's 80F and 50% RH in a confined space....this means the air is holding half as much water vapor it's capable of holding AT that temperature. If you were somehow able to raise the temperature to 90F, but allow no water vapor to enter or leave your space, the Relative Humidity would drop. This is because 90F air is capable of holding more water vapor than the 80F air can hold. Again, we didn't change the amount of water vapor in the air. So stating the RH may seem worthless, since it doesn't tell you how much actual water vapor is in the air...but the percentage of water vapor relative to the air temperature (i.e. the definition of relative humidity) is a major player in our weather.
Since RH is a percentage, it can only be in the range between 0% and 100%. As you would guess, 0% RH means that, for the given air temperature, there is no water vapor in the air (a condition than virtually never occurs naturally). At 100% RH, this means that the air is "saturated" at that air temperature, meaning it can hold no more water vapor. Depending on various factors, this can result in rain..or it can cause water to condense on slightly cooler surfaces (similar to the way the water vapor in your breath can condense on the outside of a glass of ice water). This "saturation" point is also called the dew point. The dew point is just a fancy way of asking...if we don't change the amount of water vapor in the air (just like in the example above), at what temperature would the RH be at 100%...or saturated...or its dew point.
So for another example, say it's 80F, and the weatherman says the dew point is 64F. This is the same thing as saying, if the amount of water vapor in the air doesn't change, and the temperature drops to 64F, that will be the temperature at which the relative humidity is at 100%, and the air is saturated. As you can guess, it's called the dew point, as this often occurs at night or early morning, when the air temperature has dropped (due to sunset) enough to allow water to condense on plants and grass, creating dew. If the temperature continues to drop, it will rain, since the air can no longer hold the water vapor.
So this hopefully answers your question..as the air temperature gets closer to the dew point, the relative humidity increases until it's at the dew point, at which the RH is 100%, and you may get rain or dew (water condensation) on some surfaces.
I hope this helps!