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Improving acceleration comes down to two concepts: the amount of force you put into the ground and the ability to apply that force quickly.
Newton's Third Law of Motion beautifully summarizes the basic physics of acceleration: "To every action there is an equal and opposite reaction."
Applied to running, you might say, "For every force applied to the ground there is an equal and opposite force applied back to the runner."
Strength training helps you increase force. The more you are able to push into the ground with each step, the more force the ground will push back into you, causing you to move faster.
Here are three strength-training exercises to get you started.
A staple of most athletic lifting programs, the Deadlift can help you increase force by building your overall lower-body strength. Since this is a slower movement (and Force = Mass x Acceleration), it's more important to lift heavy weight than to do a high number of reps. A general recommendation is 3-5 reps for 4-6 sets.
The Power Clean is a highly technical movement that, when done properly, allows you to generate a large amount of force. Always seek proper coaching before adding Cleans to an exercise program.
Looking back at the force equation, it is easy to see why the Clean is such a useful exercise. It can be performed with a fairly high amount of mass while you accelerate rapidly. The exercise begins to bridge the gap between strictly developing force (the Deadlift) and developing force quickly (Plyometrics). The repetitions should be kept low, in the 1-5 range, while sets can vary from 4 to 8.
Glute Ham Raise
Although not strictly about developing force, Glute Ham Raises (GHRs) can greatly benefit any athlete attempting to improve acceleration. By training the hamstrings in their entire range of motion, they can help prevent injury during sprints. GHRs also train the spinal erector muscles at multiple angles.
Keep your back straight when you're at the typical 45-degree angle. This will help propel you forward. Since maximum force is not the goal of this exercise, repetitions can be as high as 10-12 with sets in the 2-4 range. If you don't have access to a GHR machine, substitute Hamstring Dips.
In addition to these exercises, you can add plyometric work to increase the speed at which you generate force.