In SQL Server 2012, upgrading edition is actually easier than ever since the data is stored under the same directory.
Backup your data data. For default installation, the data can be found in the following directory:
C:\Program Files\Microsoft SQL Server
Have a local copy of SQL 2012 setup files as it will make the installation process much faster.
When SQL Server Installation Center launches, click on Maintenance on the left pane and select Edition Upgrade on the right.
Make sure after the Setup Support Rules shows nothing failed.
Click OK. It will run again one more time to make sure that the computer is not a domain controller. Click Next when done. You need to do this a couple times until you see the screen where you need to enter the product key.
Enter the product key, and click Next.
Pick the SQL Server instance you want to upgrade.
Click Next > until you reach the Ready to upgrade edition.
Check the summary on the Ready to upgrade edition.
If everything looks correct, click on the Upgrade button.
Once done, you should see all statuses as Succeeded in the Complete screen. Click Close.
If you want to double check, connect to the database open a New Query window and run this SQL statement:
It should display result similar to this:
Microsoft SQL Server 2012 - 11.0.2100.60 (X64) Feb 10 2012 19:39:15 Copyright (c) Microsoft Corporation Developer Edition (64-bit) on Windows NT 6.0 <X64> ...
You might also want to remove the SQL Server 2012 setup files from the local hard drive.