All Safari Users - Usually iPhone or iPad

Tap the Share button.  This is the button immediately to the right of the address bar. It looks like a box with an arrow coming out of it. 
You should see "Add to Home Screen" in the second row of buttons.  It has a big plus sign in the middle of the button and is right next to the "Add to Reading List" button.
After you tap the Add to Home Screen button, a window will appear with the name of the website, the web address and an icon for the website.  You shouldn't need to change anything, but if you want to give the website a new name, you can tap on the name field and enter anything you want.
Tap the Add button in the top-right corner of the window to complete the task.  Once you tap the button, Safari will close and you will see an icon for the website on your home screen.

All Android Users 

Launch Chrome for Android and open the website or web page you want to pin to your home screen. Tap the menu button and tap Add to homescreen. You’ll be able to enter a name for the shortcut and then Chrome will add it to your home screen.
The icon will appear on your home screen like any other app shortcut or widget, so you can drag it around and put it wherever you like. Chrome for Android loads the website as a “web app” when you tap the icon, so it will get its own entry in the app switcher and won’t have any browser interface getting in the way.
Other popular Android browsers also offer this feature. For example, Firefox for Android can do this if you tap the menu button, tap the Page option, and tap Add to Home Screen.

All Windows Mobile & Tablet

Windows 8, 8.1, and RT devices also offer a way to pin websites to your Start screen. This is obviously most useful on tablets, not on desktop PCs where you don’t want to see the Start screen. On the Windows desktop, you can pin website shortcuts to your taskbar for easier access.

First, open the modern Internet Explorer browser — that’s what you’re probably using on a tablet, anyway, as it offers the most touch-optimized experience. Navigate to the website you want to pin, pull up the app bar — for example, by right-clicking or swiping up from the bottom of your screen — and tap the star icon. Tap the pin icon, enter a name for the shortcut, and click Pin to Start. The website will appear as a tile on your Start screen. Tap the tile and the website will open in Internet Explorer

The process is similar on Windows Phone. First, open the website you want to pin in Internet Explorer. Tap the More (…) button and tap Pin to Start in the menu that appears. Windows Phone 8.1 supports live tile updates from websites that have configured the feature, just as Windows 8 does.

It would be impossible to cover all types of mobile devices and OS's in this guide. But...
If you have another type of smartphone or tablet, it probably has a similar feature to the above, too. Just open its browser and look in its menu for an option named something like “Add to home screen” or “Pin to home screen.”