That’s the ball cock, or water inlet valve. This valve is operated by the “wire” and “ball” mentioned in your question. The wire is called the float arm and the ball is called the float ...

Rafael Graça do Prado, 32, who runs a tourism boating business, has told how he spotted the large plastic ball while in the sea off Lazaro Beach near Sao Paulo, Brazil ...

The float, the flapper, or the fill valve are the most likely causes of a toilet running continuously. You can perform easy tests like monitoring water movement to help identify the cause.