My oboe instructor makes/tunes my oboe reeds. Every time he scrapes the reeds, he inserts a piece of plastic(?) into the reeds. What's that piece of plastic called? It's a bit wider than the reeds, round on one end and a bit slimmer on the other end.

The "plastic thing" is called a plaque. It can be made from plastic or metal. Plaques can also be flat or shaped.

In addition to what Canadian said,

The plaque lets your teacher scrape the tip of the reed w/out damaging the tip (though students can still take chunks out of the tip with dull knives, as can pros). It's a very useful tool. It also helps you to see if the tip is thick enough. There are too many uses for a plaque.

Hope that helps.