Migraine is a type of severe headache caused by over-dilation of the blood vessels serving the brain. Migraine pain usually localizes to one side of the head, but is often intense enough to cause dizziness and nausea throughout the body. Along with the pain, which is often intense enough to put the sufferer out of commission for the day, sensitivity to sound and light is increased manifold, turning those stimuli into sources of pain as well.
Here's more-or-less science's best idea of what happens physically during an attack: First, blood vessels serving posterior, ventral brain areas constrict much smaller than they should, due to a variety of triggers partially listed below. Note that vasoconstriction also causes many garden-variety headaches, but with migraine the constriction serves as a predecessor to what actually causes the pain. Next, nerves in the meninges, a layer of tissue surrounding the brain, sense the constriction and signal the blood vessels to dilate. The vessels then do so, but because they had constricted so much, they overcompensate and dilate much more than necessary. The throbbing, sickening, unholy migraine pain is caused by too much blood rushing too quickly through the vessels themselves, doing them additional damage.
Migraines may also cause visual distortions and hallucinations before the pain starts; this pre-effect or "aura" happens to about 20% of migraine sufferers. The blood vessels which constrict in migraine are at the back and base of the head, and feed the occipital lobes more than any other area. As it happens, the occipital lobes are responsible for vision processing, containing the primary visual cortex. When blood flow ceases to these areas, an area of neural depression spreads out from the stanched vessels, shutting down the neurons within it. Blood eventually starts flowing again as noted above, so the hallucinations taper off right as the headache pain begins to come on. See my writeup under migraine aura for more on this.
Some possible triggers for migraine are: Stress, fatigue, meat processed with nitrates, chocolate, aged cheese, MSG, abnormal hypertension, red wine or alcohol in general, menstruation, changing/irregular sleeping patterns, and withdrawal from caffeine, alcohol, or other drugs. My own reactions are usually from food, often from meats like cold cuts. Once, one of my roommates brought home a bowl of lo mein that the kitchen had spiked with MSG for a hated customer; within five minutes of beginning to eat it I was having aura effects, and ended up getting a nasty headache even though I threw it out right then.
Occasional migraine sufferers usually have to use non-prescription medication to combat their attacks, as it's hard to get to a doctor for medication when one is lying in their apartment half passed out from pain. Serious and chronic migraine sufferers use ergot derivatives (Cafergot) or triptans (Zomig) for their vasoconstrictive effects, along with hardcore analgesics for the pain. I've had success self-medicating with opioids (Tylenol 3 and Lortab) and butalbital compounds (Phrenelin). My luck with OTC medications has been much spottier, with aspirin giving minor relief and acetaminophen and ibuprofen giving no relief even at heroic doses. If some joker tells you to smoke grass during a migraine or its onset, don't do it, as cannabis is a minor vasodilator and will likely make things worse -- personal experience speaking.
Chronic sufferers can take prophylactic doses of ergot alkaloids to keep their blood vessels at a steady state. Also, type two beta blockers are known to help keep smooth muscle relaxed, and can thus steady blood vessels as well. An acute treatment available to those who know they will be having crippling attacks is a subcutaneous injection (via a cool high-pressure spray gun as seen in Pi) of sumatriptan (Imitrex), which agonizes specific serotonin receptors on the blood vessel muscle, quickly bringing it up to normal tone from distension.