How are Pets Euthanized?

Our 12 year old Jack Russell Terrier's health has been declining over the past few months, what are our options for euthanization?

I am not sure what you mean by 'options.'
   
One either euthanizes or one does not.
   
However, here are some 'tips' as it were.
   
Your dog must be calm so that your vet will be able to easily tap the vein (usually in a forepaw). If your dog is hyper or nervous your vet will want to sedate him and it is best if you agree.
   
Whether or not you choose to be with your dog or wait outside is up to you.
   
We have always opted to be there and it is over quickly and peacefully but I am not judging either way; it's a personal choice.
   
My best advice is that when the time comes do not let your dog down by wavering. Letting go is the hardest thing to do but if you let him linger it will really be for yourself, so that you can have a longer goodbye, and not for him. The end should be all about him.
   
Trust me, we had to let go of our newborn son and the last thing we wanted was for him to suffer so we could have more time with him, and we wouldn't do it to a pet, either.
   
It will be hard (I won't kid you) but this is the way of all flesh as we all know.
   
Nothing will quite prepare you for the moment if you have never done this before but try to hang on and be strong and calm for your dog. Let his last sight be of your smile to send him on his way with confidence. Show him the face he loved one last time.
   
Afterwards by all means bawl your eyes out. The vet and his staff have seen it all before and any other pet owners in the waiting room will be sympathetic (and if they are not, screw them, who cares what they think?).
   
I hope I have said something helpful. Good luck to you.