To renounce is to officially give up or turn away from. People on a diet usually renounce pizza and chocolate cake, for example.
The transitive verb renounce is a stronger, more formal way of saying that you reject or disown something. A prince who's tired of the royal life could renounce his title and become a commoner, or a senator who wanted to become an independent would have to renounce his political party ties. You'd be wise to heed the wisdom of Jean-Jacques Rousseau, who said: “To renounce liberty is to renounce being a man, to surrender the rights of humanity and even its duties.”