Handheld stick with bottoms that is widely used to play arcade games?

The handheld stick with buttons you're describing is called a joystick.

While the term "joystick" can refer to many different types of controllers, the classic arcade joystick is the one you're thinking of. It has a stick that moves in all directions and buttons for actions.

Here's a more detailed description:

* Stick: The joystick itself is usually a lever that can be moved up, down, left, right, and diagonally.

* Buttons: Arcade joysticks typically have one or more buttons, often labeled "A", "B", "C", etc.

* Base: The joystick is mounted on a base which is usually attached to the arcade cabinet.

Arcade joysticks are essential for playing many classic arcade games like Pac-Man, Donkey Kong, Street Fighter, and Mortal Kombat.