Running drills prepare and improve the body for running in many different ways, not limited to:
1) Warms key muscles, increases heart rate, opens capillaries
2) Improves reaction time off the ground
3) Mirrors running motion
4) Provides an active stretch

I thoroughly enjoy the build-up to running, moving from leg swings, to drills, to easy jogging, to the main set. I feel better when I go through this process. And honestly, the older I get the more I like it, and need it. For people just starting out, ensuring your body is ready to run via a gradual warm-up, I believe, gives you the best chance to remain injury free. Nobody starts out a workout with sprints.

Here are some of my favorites, the workhorses of the drill sets I do:
They are demonstrated by Coach Gilbert Tuhabonye. Read his story.
http://en.wikipedia.org/wiki/Gilbert_Tuhabonye

#1 – Side Steps (wait 25sec for this particular video to start, you’ll get a black screen at first)

#2 – Crossovers

#3 – Skips

#4 – Butt Kicks
#5 – High Knees
Note: For High Knees, I like to do 10 at a strong effort with full knee lift and a good, slightly forward-leaning body position, then go easy for 10 steps, and do this three times before resting and doing it again. This allows me to maintain perfect form vs. getting tired and starting to lean back as I do them, or losing knee height. Coach Tuhabonye does about 20-25 consecutively in the video.
#6 – Backwards Running

#7 – Strides