Once the update is available for your device a message will pop up on the phone. Alternatively, you can check for updates by going to the Settings application and tapping Phone update.
To carry out the update you will need to connect your phone to your computer and use the Zune Software for PC or Windows Phone 7 Connector for Mac. In some cases there will be several updates to carry out, but the final update should take you to version 7720. More information on how to update your phone is available here.
Microsoft explain the staggered roll out on Mango in this Windows Phone Blog post:
Delivering Windows Phone 7.5 simultaneously to so many phone models and carriers requires the right engineering balance. Speed is a priority—but so is quality. We’re not just delivering our new operating system but also new software supplied by individual handset makers. This “firmware” is necessary so your phone—and apps—work with all the features of Windows Phone 7.5. But it essentially means that we’re supplying not just one update, but many different ones, given the variety of Windows Phones and carriers out there to choose from.
If a problem comes to light, it’s critical that we can isolate and fix it quickly. So we’re deliberately starting out slow. This week, we’ll be making the update available to 10 percent of customers. If everything looks good, we’ll open up the spigot a bit more—to around 25 percent. We’ll hold there for one or two weeks, then quickly ramp up to 100 percent—monitoring quality the entire way. That’s how we ensure Mango arrives both quickly and in tip-top shape.
More information on the blog here.