Neuralgia means nerve pain. This nerve pain can be caused by several different things.
One possible reason for trigeminal neuralgia is a blood vessel, a small tube that carries blood from the heart, that is pushing on the trigeminal nerve. Over time, this pressure will cause the protective coat around the nerve, called a myelin sheath, to wear down. As the protective coating wears away the nerve is exposed, the blood vessel continues to pressure the trigeminal nerve and the nerve becomes very painful.
Another cause of trigeminal neuralgia is Multiple Sclerosis: a disease that affects the nerves inside the brain. Read more about MS and Trigeminal Neuralgia in this post.
Finally, trigeminal neuralgia can be caused by a physical injury to the nerves of the face. Physical injury can damage or cause pressure to be put on the facial nerves, resulting in facial pain.
All of these causes result in the same rare, on-going condition called trigeminal neuralgia, which affects the trigeminal nerve in the face and causes severe pain.