I've seen a bottle of an additive for aquarium called stress relief or conditioner or something like that. What does this actually do and is it necessary to use it? Is it especially needed when I add a new fish to the tank?
If you are referring to API Stress Coat, it is a product that is a combination of chlorine remover/chloramine neutralizer AND aloe vera to purportedly replace the slime coat on stressed fish. It is not necessary IMO to use a product to replace slime coat on fish, even for new fish.
It is usually not recommended to add a single fish to a tank that already has established fish as there is the possibility of that fish becoming targeted. You could add one fish a few hours after lights out but definitely keep an eye on the behavior of all the fish and be prepared to remove any harassed fish.
The only water conditioner you should be using is one that eliminates chlorine or neutralizes chloramine from your tap water when doing water changes.