This application for teaching assistantship is not an application for admission. All applicants for teaching assistantships must also submit an application for admission to the desired graduate program.
The online application for admission has the option to (1) check a box that says you are interested in assistantship, and (2) list at least three references to write letters or recommendation. Let your references know that you are also applying for a TA so that they can address your ability to communicate and teach a class.
Teaching assistant applicants for fall semester are encouraged to complete both assistantship and admission applications by Feb. 15, although late applications will be accepted.