Ok, this is a big question. Many people have referred to forcing an aspect ratio, which is a good solution to consider. However for many games this is not appropriate. For many game designs, you want the game to feel like it's native to the device/screen it's running on, and you want to use every square inch of screen real-estate you can. To do this, you must build and test every system in your game for different sized and shaped screens. It's a very similar problem to building apps and websites for multiple devices.
Screen Size
The screen size can be retrieved programmatically using the DPI and the resolution with a statement such as:
Resolution res = Screen.currentResolution;
float screenSquareInches = res.width * res.height / Screen.dpi;
For the content on the screen that is viewed through the camera, there's only one variable that needs to be scaled, and that's the size of the camera (or how much stuff in the game world that will be squeezed onto the screen). You can control the size of the camera with the field of view for 3D games or the orthographic size for 2D games. Only testing will say what equation supports the widest range of screen sizes, but something along these lines could serve well:
// this should be a tweakable number, so you can fine tune it to your needs. It represents the screen size at which the camera size should not be scaled at all
float expectedScreenArea = 20;
// this number should also be tweakable, which declares how much smaller/larger text can get depending on the screen size. Generally larger screens are viewed from farther away, so scaling should be slight
float cameraScalingRate = 0.25f;
// multiply this number by the default/base camera size you have picked, whether that's field of view or orthographic size
float cameraSizeMultiplier = 1 + (screenSquareInches / expectedScreenArea - 1) * cameraScalingRate;
Ok, that should get you most of the way on scaling the game world, but what about UI elements, things like text, HUD, and on-screen controls? Well, for elements like text, the above equation should work decently well, though you might want a different value for the scaling rate. For on screen controls, you might want a more full transformation, such as joysticks moving to the bottom corners of the screen on a tablet rather than centered on the sides, or if it's not a touch screen device, remove on-screen controls altogether. That's outside the scope of this question.
Screen Aspect Ratio
The aspect ratio of the screen (how wide vs. tall is it) is also important and needs it's own handling. Generally the best way to do this is to freeze the vertical height of the screen, so that as the screen gets wider, elements on the screen are not scaled, but as the screen gets taller, elements on the screen scale proportionally to how much taller the screen grows. This is already the default behavior for cameras, but for UI elements you have to set it as such. This is relatively easy if you are using the new UI system with canvas'. On a CanvasScalar component, there's a property that controls how the elements are scaled depending on the aspect ratio. Change this to match width or height, and have it match height.
For Further Consideration
With supporting different screen sizes and shapes there's more to consider than just making everything fit. You also need to make sure you game plays well on the different screens. This ranges from things as subtle as not making the player feel claustrophobic on a small screen to making sure enemies can't shoot at the player if they're off screen to making sure you update boundaries that keep elements on screen.
Another thing to consider if this is a PC game is that someone might plug their computer into a different screen or change the resolution on the fly. With this in mind, it can be good to rerun the above equations as needed so a player doesn't have to exit the game to get things lining up right again.