The PS1 gamepad controller has the same button and analog stick layout as a PS2 dualshock 2 gamepad controller. If you put them side by side, they look almost identical to each other, the only difference would the PS2 controller is a black color. The PS1 controller has the same controller port and pinout as a PS2 controller, so technically if you plug it into the controller port on your PS2 console it will work. Using a PS1 controller on your PS2 console will certainly allow you to use it to play PS1 games. If you decide to play PS2 games using the PS1controller you have plugged in, most games will allow it to function with the game, so it will work. However, some games don't have backwards compatibility with PS1 controllers and will only support PS2 controllers, since the PS1 controller doesn't support dualshock 2 vibration technology.
Instructions on How to connect PS1 controller to a PS2 console
1) Turn off your PS2 console by holding down the 'Reset' button for 3 seconds. This will put the console into standby mode.
2) Get your PS1 controller you want to connect into your PlayStation 2 console. Now you will need to find the PlayStation controller's plug. It is a chunky square bit of plastic that is found at the end of the controller cord. This plug connects into the controller port found on the PS2 console
2) Your PS2 console will have two controller ports integrated into the system, which will allow you directly connect a maximum of 2 controllers. Decide which controller port you want to connect your PS1 controller to. If player 1 is going to use it, connect it into controller port 1, if player 2 is going to use it, connect it into controller port 2. So connect your PS1 controller to a controller port of your choice, by inserting the PlayStation controller's plug into the port.
3) Once the controller is connected into a controller port, press the 'Reset' button on the PS2 console and it will power itself back on. If the PS1 controller is connected properly, the red color analog led will light up.