Upgrading SQL Server 2008 R2 Standard edition to SQL Server 2008 R2 Enterprise edition what is the process and the impact?
The process is simply to launch SQL Server setup from your DVD or ISO, go to the Maintenance tab, and perform an "Edition Upgrade" (please ignore the fact that this screen shot is from SQL Server 2014):


  


You'll need to enter a different product key.

Please also make sure you install the latest service pack (SP3; 10.50.6000).

As for impact, you shouldn't expect any negative impact at all, except that perhaps you will need to bounce the service as a part of installation.

More info on edition upgrades.

If this doesn't work out, you could always install a brand new instance of SQL Server with a different instance name, then backup and restore your user databases (and copy linked servers, job, and logins). But that really should be a last resort - if you have an issue with an edition upgrade, come back with a more specific question.