I have to deal with euthanasia on a daily basis at work, and it never gets easier. I don't think it is ever an easy decision for anyone.
The drug used to euthanize is usually a very powerful barbituate. It is basically an anesthetic drug in such a powerful concentration and dose that instead of just becoming sleepy and going unconscious, it progresses to the death stage.
Euthanasia for our rodents is different than other pets, because you can't inject into a vein like you would on a dog or cat.
The two common methods to get this drug into the system in a rodent are IP, intraperitonal which is injecting into the abdominal cavity and IC, intracardiac which is an injection right into the heart.
When doing IP the drug will take longer to absorb, and it can be awhile before sufficient drug is absorbed into the body to cause the euthanasia. I have read up to 15 minutes, but from personal experience it is actually much more rapid, within 5 or so minutes.
When doing IC the drug is going directly into the heart which leads to rapid euthanasia.
Needle pokes are painful, especially since our rats are so tiny. For this reason it is often recommended to gas the rat down with an inhalant anesthetic such as isoflurane first. They breathe in the gas and fall asleep in just a couple of minutes. When they are asleep the injection (usually IC) can be done and they won't feel a thing. IC injections should never ever be done on an awake rodent. That is inhumane.
If gas absolutely is not an option, doing an IP injection and then covering the animal with a blanket and putting it in a quiet place to fall asleep is not inhumane. I don't prefer it, and we all love our pets dearly and would prefer them to pass quickly, but they are not in any pain after the initial needle poke.
I hope this helps you understand what happens.