Every controller is different, and every game has different controls. Regardless, there are button mappings that are fairly universal.
In many platform games, the title screen will ask that you press start to begin. And on almost every controller, there is a button correspondingly labeled “Start”. This button also serves to pause the game.
The cross-shaped structure is known as a D-pad (Digital or Directional Pad). In almost every 2D platformer, the directions you can travel (left and right) are mapped directly to the directions of the arms of the D-pad. Usually, the up and down directions either do nothing or allow you to look up, look down, or crouch.
Actions other than movement are performed by pressing a button. Jumping is the simplest and most important action, and is therefore usually assigned to the “primary” button. No button is officially dedicated the primary button, but it is usually pretty obvious. If the buttons follow a letter naming scheme, it is usually the A button. On controllers based on the Super Nintendo controller, which features a cross-shaped pattern of buttons that parallels the D-pad, it is usually the bottom button. Sometimes, such as in Sonic the Hedgehog, the jump button is all you will need to be concerned with, but many games have other actions. Mario runs with B, and many characters attack. These secondary actions are usually performed by pressing a button easily reachable with the thumb without moving it from the jump button. Some games have even more actions, and there’s not usually any reason behind which button they are assigned to. Don’t be afraid to experiment, or consult the instruction manual. Soon, you will have memorized which buttons perform which action.
In 3D games, a much greater range of motion is required than can be provided by the D-pad. In these games, an analog stick is used for movement. It follows the same principles as the D-pad, but can move 360 degrees.
Camera control is one of the most difficult aspects of 3D platformers. Without a clear view of your surroundings, it is hard to do anything, so it is one of the most important, as well. In many older 3D platformers, the shoulder buttons allow you to rotate the camera horizontally around your character in two directions. Rarely, these actions are assigned to some other button. In most newer games, the second analog stick controls the camera. Pressing left and right will perform the same action as the shoulder buttons, while pressing up and down will either zoom in and out or rotate vertically about your character. Often, there is also a button designated to centering the camera behind your character.