What is a Flipboard magazine? Glad you asked. It's a way to get your news delivered to you on your tablet or phone.
First of all, you will need an Apple iPad, iPhone or the Android equivalent. Flipboard doesn't work on a PC or even a Mac. Use your Apple or Android app store to download and install Flipboard on your device (some Android devices may already have it loaded by default). Enjoy adding dozens of high-quality news sources to your Flipboard. The first of which, of course are listed right on this page.
Update: Flipboard now lets you preview magazine contents inside some modern browsers. Safari for Mac works and Chrome seems to work on all platforms. Firefox should work (don't have an installation here to test it). If you are using Internet Explorer, all I can ask is, "why?" Click on the magazine that interests you below and see if it works for you.