Recovery after a hard or very long workout during which muscles fibers suffer the microtears the stimulate the bodies healing process to strengthen the tissue, is important. If an athlete is very sore, some people believe it's better if they rest entirely until the soreness has abated, and this makes good sense, because the degree of soreness may be related to the amount of microtearing that has occurred during a hard workout. If the soreness is miles, an athlete may instead choose to do what is called "active" recovery, which is they may choose to do a very easy run instead of resting completely on an off day.
Active recovery is a popular concept in training for runners today, and yet many people will call any run of any intensity a "recovery run" if it's done the day after a hard workout, or even later the same day. So, many people will go out for an easy run, then run progressive splits escalating to a tempo pace or better in the final mile. Or they may just do the whole run at a moderate to high-intensity effort. It's natural for people to run faster if they feel like it.
If the idea of active recovery is correct, however, it's important to keep a recovery run at a "recovery" pace. Recovery pace turns out to be about as slow as you can stand to run, but not so slow that your stride is abnormal or choppy. The idea is to keep a good range of motion and your standard running form, to get the heart rate up just a little and the blood circulating in those torn down muscles from the previous hard workout, but not to run at an intensity that would further stress or tear down muscle fibers.
An obvious difficulty with this is that some people just are not in good enough condition to accomplish those goals when they early in training. People early in training should probably use passive recovery until their condition improves and their muscles strengthen a bit, because running at even slow paces may hamper more than speed/improve recovery for them.
For folks who have been running for awhile, though, active recovery may help speed recovery and further develop or maintain aerobic benefits gained in other training. This article by Matt Fitzgerald on active.com suggests a plausible reason for the apparent fitness benefits of active recovery. According to Fitzgerald, many of the fitness benefits of exercise accrue essentially in proportion to the amount of activity performed after the onset of fatigue. An active recovery run, after a long or difficult workout would be entirely in a fatigued state, so theoretically would provide fitness benefits as a result.
Active recovery. All the cool kids are doing it.