Why didn't my iPhone 4 update to iOS 6?

A few days ago I updated my iPhone 4 to iOS 6 (that was iOS 4.2.6) on my cousins computer (on iTunes of course) but it didn't update at all. It's still at iOS 4.2.6 and I want to know why it didn't work. And how will it work next time?

Go on your phone to settings then to general, then click on software update at the top. It will check for any available updates, and if none are, it will tell you your software is up to date, and which software version you are using. (make sure you are connected to the internet when doing this.) If it says your software is up to date and it says you are using 4.2.6 you might want to consider taking it to the apple store

You should never connect your iPhone to someone else's computer or iTunes. Chances are that you will mess up the settings big time, because you are mixing two people's accounts and identities. Stick to your own computer. Connect to your own computer, or update via iCloud.