Adminstrate is a superfluous word formed via backformation from the noun administration. It always bears replacement with the shorter, older administer, which means (1) to have charge of, (2) to give or apply in a formal way, (3) to dispense, or (4) to manage. Administrate has no definitions of its own.
Example
In these examples, administrate could be replaced with administer:
They dohave a freak show to administrate, after all. [Slate]
Crawford County Commissioners at their meeting Friday unanimously approved the hiring of an employee to administrate a $49,000 grant to help educate pregnant teens. [Pittsburgh Morning Sun]
Advocates of such drilling suggest issuing tough but clear new rules, and establishing a mechanism to efficiently administrate the applications for drilling permit requests that are submitted. [NPR]
With GaGa now clean and her career taking off, Joe helped her set up companies to administrate her publishing, merchandising and Haus Of GaGa activities. [The Sun]