If you are talking about groups for organizing you contacts, the only way to do this in the Contacts app is using an iCloud account. The Groups have to be set up and managed using the icloud.com site from a computer's web browser.
Unless you have a Mac; in which case you can do all the managing in the OS X version of Contacts and sync with your iOS devices through iCloud.
None of the other contact syncing options can handle groups. Apparently it's a limitation of the syncing methods used; some thing Apple can bypass when using their own services.
If you are talking about mailing groups, many members have found MailShot does what they need. It creates special entries in the Contacts app that act like mail groups.