These rubber parts fit between the plastic buttons you press, & the sensors on the circuit board inside the controller.
The original parts can wear/split, which can stop the buttons from working properly. Simply replace any worn identical parts with these.

Will fit PS1, PS2 & presumably PS3(I haven't tried this myself, but seems likely it uses the same parts) controllers.

Quite easy to replace, the only difficulty is disassembling & re-assembling the controller.

Click on picture, to see enlarged photo with text labels(to show which buttons each rubber part is for).