Relative Humidity is the ratio of of the quantity of water vapor present to the quantity of water vapor in saturated air at the same temperature.

Let's take that one step at a time. Firstly, understand this: There is a certain amount of water vapor in the air. That amount ranges from zero (see: winter in New England) to a maximum value. When that maximum value is reached, the air is said to be saturated with water. Any more water vapor, and the whole of earth's atmosphere would just sort of THWOMP into water (water-nine, anyone?). Actually that last part isn't true. Just look up super saturation for that. Keep in mind, also, that the saturation point varies depending on several factors, such as temperature and pressure.
Now that we've set a minimum and maximum amount of water vapor that can be in the air, we have a scale on which to measure relative humidity. Just compare the actual humidity measurement to the maximum humidity, and you'll get your ratio or percentage. It's basically just a Holy-fuck-how-much-worse-can-it-get measurement.

Specific Humidity: In a mixture of water vapor and air, the weight of water vapor per unit weight of dry air. Also known as humidity ratio.

Instead of a min-max scale, as above, this is simply a comparison of the weight of the two components of "air": the water vapor and the actual "dry air". Let's say you have a metric ton of everyday, over-the-counter air. A portion of that - say 1/4 - is actually composed of water vapor. The other 3/4 is "dry air". Weigh the water, weigh the air (good luck), and there's your nice ratio.