The thoraco-abdominal diaphragm separates the chest cavity from the abdomen. It attaches to the xiphoid process of the ribcage and to the vertebral column, and essentially separates the top half of your body from the lower half. It is the main muscle used in respiration (breathing).
When the diaphragm is at rest, it curves upwards in the middle. When you inhale, the diaphragm contracts, pulling its center down, and consequently expanding the chest cavity. The lungs 'stick' to the inside of the chest cavity, so when it expands, so do they, sucking in air. At the same time this stimulates the stomach and liver to help digestion along. When the diaphragm relaxes it squeezes the air back out of your lungs. This action alone is enough to keep you well oxygenated, although other muscles (such as the intercostals) will become involved if you need to breath more heavily, or speak above a whisper. The diaphragm is innervated by the phrenic nerve.
Hiccups are the result of muscle spasms in this diaphragm.