iOS 6 represents a major update to Apple's mobile operating system for the iPhone and iPad, and as long as your device isn't more than around two years old you'll be able to upgrade it to get some or all of the new iOS 6 features.
How to install iOS 6
Of all Apple's iOS devices made since 2010, it's really only the original iPad which didn't get an invite to the iOS 6 party. Here's a full list of what's compatible and what isn't. See also: Which iOS 6 features will my device get?
iOS 6 compatible iPhones: iPhone 3GS, iPhone 4, iPhone 4S, iPhone 5 (ships with iOS 6)
iOS 6 compatible iPads: iPad 2, new iPad (third-generation)
iOS 6 compatible iPod touches: iPod touch (4th-generation), iPod touch (5th-generation, ships with iOS 6)
The following devices can't be upgraded to iOS 6:
Original iPhone, iPhone 3G, original iPod touch, iPod touch 2nd-gen, iPod touch 3rd-gen, original iPad.
There are two ways to upgrade: over the air (via Wi-Fi) or through iTunes. If your device is running iOS 5 or later, you'll be able to update to iOS 6 without needing a computer since iOS 5 added support for over-the-air updates. If your firmware is iOS 4.x you'll have to connect to iTunes and perform the upgrade. Take a look at Can I get iOS 6 on my iPhone 4S? too.
Over the air iOS 6 upgrade
You should see a pop-up notification telling you that a software update is available for your device, but if not, tap Settings, General, Software Update. Your device will check if a new firmware is available and you can then follow the instructions on screen. Visit Should I upgrade to iPhone 5 from iPhone 4S?
When prompted, tap Download to download the iOS 6 update. Once this is finished, tap Install to upgrade iOS.
Before you perform the upgrade to iOS 6, though, you should back up your device either via iCloud or iTunes. It's unlikely anything will go wrong, but if it does, you could lose any data that's changed since your last backup such as new contacts, documents, photos, videos and more.
As the iOS 6 update will be several hundred megabytes, you should also plug in your iPhone, touch or iPad using its mains adaptor, or have at least 50% battery power.
Installing iOS 6 via iTunes
If you're running a version of iOS before 5.0, you'll need to connect your device to a computer running iTunes. You should update iTunes to the latest version (currently 10.7) before connecting your iPhone, iPad or iPod touch. To do this, launch iTunes and from the Help menu, click Check for Updates.
Once you have iTunes 10.7, connect your device and it should display a message saying a software update is available. If you don't see this, simply click on your iPhone, iPad or iPod touch in the Devices list on the left and then click the Check for Update button.
You'll be prompted to download and install the update - make sure you don't disconnect your device before the upgrade process is finished.
Whether updating over the air or via iTunes, your iOS device will restart after installing iOS 6.