I'm deeply sorry you have to make this decision for your companion, but I commend you for being selfless enough to do what's right for your bird.
Your friend is first put to sleep with an anesthesia, a little cone might be held over his head or he might be placed in a container with the gas piped in until he's sleeping. It's a very gentle sleep. Once he's sleeping, the drug necessary to stop his heart is injected. He does not feel anything.
Some vets may use different procedures so it's important to ask them beforehand and let them know all of your concerns. Instead of anesthesia they may inject a barbituate to induce sleep and then follow with the injection of a heart stopping drug.
Again, I'm truly sorry for how hard this must be for you.
What kind of bird have you got?
How old is he ? How long has he been with you?
Tell me everything you can about him, what he eats, what his day is like - ok?