The world is your oyster = you can do anything you like in the world

An oyster contains a pearl. Simply open it up and the pearl is yours. The common expression is "the world is my oyster", ie. the world has something valuable for me, and I can simply take it.

one's oyster? If you mean "the world is one's oyster" then it means the chance to do something is open to that someone. It also implies that that someone has to seize the opportunity if he/she wants to achieve something.