When you are ready to have your pet put to sleep, you can either take him or her to the veterinarian's office or find a vet who will come to your home. Many areas have mobile vets; ask your veterinarian to recommend one if this is the option you'd like to choose.
The vet will probably have an assistant present. One of the assistant's jobs is to help hold the pet, and maybe also to shave off a little patch of fur in order to let the vet see exactly what she's doing.
You can choose whether or not you want to stay with your pet. When the time comes the vet will either put in a catheter, or give the pet a shot, of the same drug they use in surgery for anesthesia. The dosage is much higher, effectively an overdose. What happens to the pet is that he or she suddenly gets very sleepy. If you have ever had surgery and can remember that fast sinking-into-sleep sensation, then you know how this might feel. Because it is such a large dose, the pet feels him or herself fall asleep, and then is gone. It happens very quickly: often in just a matter of seconds.
It is so swift, and so painless, that despite their sadness the people present often feel an immediate sense of peace.