The word rate is related to "ratio," which is a way for us to express the relationship between two or more quantities. Ratios can be written as 'a to b', 'a/b', and 'a:b'.
Unit rate, then, is a ratio that describes the relationship between a measurement of one unit compared to that of another. Miles per hour, for instance, is a common unit rate. It describes the distance traveled (miles) for every single unit of time (hours). Pressure is another example of a unit rate as it describes the amount of weight applied per every unit of space, ex. 15 lb. per square inch.
The important thing to remember about unit rate, though, is that the denominator (the second quantity being compared) must be equal to one. An easy way to remember this is to think of how we commonly measure speed; road signs display speeds as 45 mph or 100 kph. This expression would be much less effective if we had to continue to reduce the fraction until the denominator was 1.
Unit rate is everywhere. We use it to express wages, for instance. Look around you and you'll notice how much unit rate affects our everyday lives.