Tap the 'Share' button on the browser's toolbar - that's the rectangle with an arrow pointing upward. This is located on the bar at the top of the screen on an iPad, and on the bar at the bottom of the screen on an iPhone or iPod Touch.
Tap the 'Add to Home Screen' icon in the share menu.
You will be asked to name the shortcut. Other browsers, such as Chrome for iOS, don't offer this feature - only Apple's own Safari browser is allowed to do this.