In all states except one, LLCs are divided into two categories--those managed by the members and those managed by managers appointed by the members (aka "manager-managed"). In a quest to offer LLCs the face of a corporation, Tennessee goes a different route. Their LLCs are either member managed or governor managed. Further, in a LLC with governors, the governors may appoint managers who serve under them in much the same capacity as officers server under a corporate board of directors. For more information on the particulars, see "New Management Structures" section of this article.
I don't really see what this statutory frameworks adds to Tennessee LLCs; however, the novel framework does create traps for the unwary. The managers of LLCs in other jurisdictions act like officers of a corporation. Further, in a manager-managed LLC, the members have a roll very similar to the board of directors of a corporation. So what's the point to LLC governors Tennessee? In the grand scheme of things, it's not that big of a deal. Governor-managed Tennessee LLCs operate exactly the same as member-managed LLCs in the other 49 states (except for the fact that they can named managers who report to them). For Tennessee LLCs wishing managers, they are required by statute to create an intermediate body of "governors" in addition to naming managers. This necessitates additional complexity in the operating agreement. However, one option to simplify governance for Tennessee LLCs with managers is to name all governors additionally as managers. The net effect of such an arrangement is that the body of LLC governors = the body of LLC managers. Although the operating agreement still requires extra provisions to address the issue of governors, I believe it's a relatively easy drafting issue to handle.
Here is a link to the Tennessee statutes. Go to Title 48 (Corporations and Associations) and see Chapter 201, et alia for the Tennessee LLC Act.