The two main areas of failure are the fill valve, and the water supply line (WSL).
Fill valve: in the old days, this was the device attached to the big copper ball that floated on top of the water in the water supply tank, and traveled down as the toilet was flushed.
Other types of fill valves, of which the Fluidmaster is the most noted, have generally replaced the ball. It is held in place by a lock nut screwed into the fill valve stem, which passes through the bottom left-hand side of the water supply tank. The coupling nut is generally plastic, and secures the WSL to the fill valve. The WSL is connected at the wall below the toilet by an angle stop.
The WSL can be plastic or braided metal.