I have a full Server license but is it only Windows 2008 Server Standard Edition.
Can I upgrade from Windows 2008 Server to R2 for free?
Is R2 a different product? Can I just upgrade it with a download?
Depending on the licensing you have, the R2 releases are a separate product from Microsoft. If you have a retail copy of 2008, you will need to purchase the upgrade. If you are in the Microsoft licensing program talk to your rep as you may be eligible for an upgrade.
R2 is a different product, it came out a year later. Windows 2008 server is a valid upgrade path to R2 (as described on the Microsoft update website). You will not be able to upgrade with just a download.
If you have Software Assurance on your license, then you can. If not, then no. See http://www.microsoft.com/licensing/software-assurance/new-version-rights.aspx.