How to remove ticks safely and correctly
Ticks, especially hard ticks, are tricky to remove. All ticks bury their heads under the host's skin, but hard ticks secrete a "cement" that firmly locks them into the host. With care and patience, you can remove the tick yourself, or you may opt to have your veterinarian do it. 

To remove the tick, use thin-tipped long tweezers to grasp the tick as close to the animal's skin as possible. Very slowly and gently, with continuous pressure, pull the tick out from under the skin. This is a delicate operation-if you pull too fast or are jerky, you may leave the tick's head (or part of its head) under your pet's skin.

Do NOT use a match or caustic materials to try to smother the tick, burn the tick or get the tick to 'back out'. This doesn't work, it can actually cause the tick to "break up" and may cause the tick to regurgitate more saliva (and potential pathogens) into the skin.  

If You Leave Some Behind...
If you do break off part of the tick's head, you will see small dark spots under your pet's skin where the tick was. You can try to tease the remaining pieces out if your pet is especially mellow, but it's best to leave them alone. The area will form a small pimple and then fall off. If your pet is very uncomfortable, though, take him to the veterinarian to have the area attended to.

Keep in mind that many of the pathogens that can infect your pet can infect you, too. You should wear gloves when you remove the tick -- especially if you need to handle the tick. After you've performed the removal, always wash your hands thoroughly.

And finally .... 
Talk to your vet about effective tick control (spray, powder, spot-on, or collar) for your pet.  

Check pet daily, especially in the spring when ticks are most common.

Update email ~ May 2005
"....I noticed that you advise people to pull the ticks out slowly. However this has never worked on my own cat as the head almost always is left in the skin. I have found twisting the tick clockwise with a pair of small tweezers works the best and means the ticks are still complete."

The above advice was emailed to me by Iseabal Macarthur from the Western Isles Scotland for which I am extremely grateful - John