There are two different explanations to describe what acupuncture is. There is what we call the TCM or Traditional Chinese Medicine definition and then there is the Western Medicine version.
Here is the TCM definition: Your body is made up of energy called Qi (pronounced "chee"). The Qi runs in channels we called meridians (like highways) all over the body. When there is an imbalance (the area is too weak or stuck) in the qi, you get sick or feel pain. The acupuncture needles get put in specific spots (called acupuncture points) to help move the energy and thus creating more balance to let the body heal itself.
This definition is actually similar to the theory in how we believe chiropractic works. When we adjust the fixations in the spine, we remove interference in the nervous system and let the body's innate intelligence take over and heal the body.
Now, the Western Medicine explanation of acupuncture describes it this way: Acupuncture works in a manner called "gate control theory". By stimulating peripheral nerves with the needles, we shut the "gate" on pain signals traveling through the spinal cord and thus to the brain. This also changes the neurotransmitters released in the brain and affects blood flow to areas involved in sensing pain.
In reality, I don't think we really understand how it exactly works. The western medicine description really doesn't explain how I can put a small needle in the side of the pinkie finger and get a migraine headache to go away! So, for now, we will just have to accept the fact that acupuncture is a very old healing art that really does work.