Collar inserts or collar stays, tabs or even collar stiffeners are the same thing. Some shirts have built in collar stiffeners in them made out of plastic (these are usually found in dress or work shirts instead of button down shirts which are more casual in nature). I prefer those that have inserts which you can remove and use on other shirts.
The bad thing about those built in ones is that they only stiffen up the collars when the shirt is fairly new . After a about a year's of washing, whether by hand or by machine, these built in inserts would actually become flimsy or bent out of shape or even wear out the cotton in the collar. Or they could suffer if you use too much heat during ironing. With removable inserts you actually get the choice of removing them when washing the shirt (which prolongs the shirt's life as the insert does not rub against the cloth) or when ironing the shirt.
