Initially many companies created more than one website, each of which was suitable for a different receiving device. Now there are so many different devices and sizes , creating different sites is no longer practical and probably near to impossible.
Responsive Web Design
Responsive Web Design (the term was coined by Ethan Marcotte in an article for A List Apart) allows you to create web pages that respond to the environment/device in which they are viewed.
This is achieved by creating Media Queries that are part of your CSS.
Media Queries allow you to specify specific criteria (size, orientation, color, aspect-ratio and more) that match the viewing device, a desktop/laptop, ipad, cell-phone, tv etc.
For Media Queries you need to create a series of CSS Styles based on the main design for your site.
You need to create a different set of styles for each type of device.
Testing the Site
You can initially test your page by just resizing the browser window but you will need to test on a variety of devices before you go live.