You should have a very good base of running down before speed training. You should know what type of pace you are comfortable running (Ex: 10-11 minutes per mile). Then you should know what is a reasonable "challenging" pace you can run and for how long (ex: A quarter mile in 2 min.). Once you have a pretty good understanding of how your body reacts at these speeds, doing a few weeks of speed workouts a couple times per week can help you become significantly faster.
Following are some of the most common ways to work on your speed (remember to warm up and cool down):
Intervals - (usually done on a track or well-marked, familiar course) During interval training, you will run for a set distance while maintaining a certain pace, then you will jog or walk comfortably for an equal or longer period of time (ex: Run a quarter mile in 2 min. and then jog a quarter mile in 10-11 min.)
Fartlek - (usually done on the road or on a trail) Not much different than interval training, but less planned and more playful (ex: Run to the tree up ahead at your fastest pace and then jog or walk until ready to race to your next landmark, like a sign or a light post)
Tempo Runs - Tempo runs are longer distances. After you are comfortable jogging for, say, 2 miles without stopping, try running a mile at a faster pace between warm up and cool down.