Button Layout
Typically a gamepad is set up such that direction is handled by the left hand, and actions are handled by the right.
Left Side/Directional
Early versions of the gamepad featured a D-pad, a 4-way directional button. Most current gamepads have substituted an analog joystick in its place (though some have both).
Right Side/Action
The set of action buttons has increased from 2 (NES) to 4 (although the first Xbox had 6). Additional action buttons have been added in the form of "shoulder" buttons (buttons mounted on the controller facing the away from the player) and analog triggers.

Other Buttons
Most gamepads also have a Start and Menu/Select buttons that is not generally used for gameplay. The Start button is frequently used to pause the game and/or bring up a menu.
Additional Features
- Force-feedback - allow the player to feel vibrations for certain actions in game
- Wireless - not required to be plugged into the game machine
- Peripheral Attachment - can attach memory cards, keyboards, etc directly to the gamepad
- Motion Sensor - capable of detecting the player's movements
- Music gamepads (Rockband, Guitar Hero) - additional button types (whammy bar, strum bar)
- Pointer (Duck Hunt Gun, Wii) - able to detect the facing of the controller in relation to the playing screen.