You'll go in the room and the vet will confirm your wishes. You'll be asked to sign a consent form to consent to euthanasia and also what you'd like to do with his body afterwards. You can either take him home for a burial, they can arrange a cremation for you, either where the crematorium scatter the ashes for you, or he can be individually cremated and his ashes returned to you, either in a scatter box for you to scatter or bury somewhere, or in a casket or urn for you to keep. They'll discuss these options with you. Alternatively you can arrange your own cremation via a pet crematorium.
The procedure itself is usually quite quick. They will need access to a vein in his leg so will clip a small patch of hair from his leg if he hasn't already had that done for the investigations. They will gain IV access either with an IV catheter, or with the needle and then the drug (it's essentially an overdose of anaesthetic) will be administered directly into the vein. They'll probably have a nurse to assist them in gaining IV access. You can be present, you can hold him and stroke him as he goes off, if you want to of course. If you don't want to be present, you don't have to be, he can be held and fussed by the nurses instead. You will be offered some time with him afterwards, again, if you wish.
He will literally just go off to sleep.
They'll pre warn you, but it's worth being aware that they can grasp and/or vocalise after the injection. It doesn't happen all the time, I'd say probably about 1 in every 10-15 euthanasias that I see do this gasping and/or vocalising. They are completely unconscious when this happens and are unaware of anything that is going on. It's just a primal reflex caused by the brain not getting oxygen after the heart has stopped.
If your dog's temperament isn't great at the vets, you could discuss the use of a sedative with them. There are pros (mainly that the animal is pretty zonked!) and cons (it can make it harder to get IV access and it also makes the process last longer, the sedative takes time to take effect) but you can discuss this with your vet.
I'm sorry that you're going through this, it's the hardest thing, but equally we're lucky that we have the option of not allowing our pets to suffer at the end.