Why do headaches hurt?

Information about touch, pain, temperature, and vibration in the head and neck is sent to the brain by the trigeminal nerve, one of 12 pairs of cranial nerves that start at the base of the brain.

The trigeminal nerve has three branches that conduct the sensations from the scalp: the blood vessels inside and outside of the skull, the lining around the brain (the meninges), and the face(mouth, neck, ears, eyes, and throat).

Brain tissue itself lacks pain-sensitive nerves and does not feel pain. Headaches occur when pain-sensitive nerve endings called nociceptors react to headache triggers (such as stress, certain foods or odors, or use of medicines) and send messages through the trigeminal nerve to the thalamus, the brain's "relay station" for pain sensation from all over the body. The thalamus controls the body's sensitivity to light and noise and sends messages to parts of the brain that manage awareness of pain and emotional response to it. Other parts of the brain may also be part of the process, causing nausea, vomiting, diarrhea, trouble concentrating, and other neurological symptoms.

This answer is based on source information from the National Institute of Neurological Disorders and Stroke.

Many of the structures around the brain and neck are sensitive to pain and various abnormalities can cause them to hurt.  Interestingly, the brain is not sensitive to pain.  The pain experienced with headache is shaped by which of these many sensitive structures are activated.

Surprisingly, the tissue of the brain itself does not have any pain sensitive nerve fibers and therefore is not capable of giving you pain. However, there are several areas around the brain including blood vessels, the membrane that covers the brain, as well as the nerves that supply the scalp, sinuses, and face that do contain vast amounts of pain sensitive nerve fibers, and therefore cause the pain that is experienced in a headache.  

During a headache, these pain sensitive areas are stimulated by stress, muscle tension, and dilation of the blood vessels. These triggers then lead to the release neuropeptides which are chemicals that transmit the message to the brain, "I hurt."