I am the executive director and due to a staff complaint I am not aware of, I have been placed on administrative leave until further investigation. Don't I have a right to be informed of the full nature of the complaint?

You have those rights given to you by the company. So what are the company rules for an investigation of this type. I would assume you would be questioned as to the complaint?

I would hire a employment attorney at this point, since it appears there must be a serous complaint.

Most of the US goes by employment at will. Which means that an employer can terminate you at any time for any reason. The company may have policies for dealing with employee complaints. They have to follow that policy. As an Executive Director you should be aware of those policies.

We can't tell you what rights you have because we don't know what the company policy is.