People tend to take their breathing for granted--it's just something that you do. If I were to ask you to take a breath, you could consciously control your breathing, but as soon as your mind began to wander, your body would take over for you again. (Which is good news for you, since if you stop breathing, you'll die!) But what really happens when you breathe?
Well, if you think back to your high school biology class, you'll recall that your lungs are like a bellows in the way they draw air into your body and exhale it out again. But your lungs are useless without your diaphragm muscle, which does the pushing and pulling on your lungs to make them work. So when you take a breath, your brain sends an electrical impulse through your nervous system to your diaphragm muscle, telling it to do its thing, but what is "its thing?"
When the order to breathe arrives at your diaphragm muscle, the nerve endings that surround the muscle are triggered and they release chemical signals for your diaphragm. These chemical signals consist of acetylcholine molecules, which are released from transmitter sites in the nerve endings. The acetylcholine molecules bind to receptor sites on the individual fibers of your diaphragm muscle. (A transmitter/receptor pairing is known as a synapse.)
When enough of these chemical signals are received by your muscle fibers, they will stimulate your entire diaphragm to contract and then relax. During relaxation, the acetylcholine molecules bound to receptor sites will break down and vacate the receptor sites they occupied, so that the contraction/relaxation cycle can start again. This happens every time you take a breath.
(If this watered-down explanation of breathing wasn't enough for you, then check out a more-detailed explanation of the mechanism of human respiration.)