If you've just installed, and it says it's expired, or say it has any amount of time less than a year, what has happened is the install has found an old expired product key, and decided to use that instead of prompting you for a new one. You need to tell it the new one:

Open Norton Protection Center. Choose the yellow button in the lower right corner. It might say 'Express Renewal' or something like that.

Input your product key right there at the bottom.

Finish the activation, and it should be showing a year until it expires.

Note that if you've purchased a product that can be installed on multiple computers, like Norton Internet Security, the clock starts ticking on the first install for all computers. If you install it two months later on a second computer, you only get ten months on that computer.