Hi kristenflowers, if you're asking about how to apply for positions outside of your home department, the following advice might be useful.
If you haven't already, visit the website of the department you're interested in and see if they have any information. If there are existing opportunities, you may be able to find details about what they entail, how to apply, the compensation, etc. You can then e-mail the contact person (if listed) and inquire as to whether applicants from outside departments would be considered. If you don't find the info you're looking for, e-mail the head of the department.
In either case, introduce yourself and state the reason for which you are contacting him/her. Briefly describe your background/qualifications, and make it clear why you are interested in working for the department in question, and in what kind of position you are interested (e.g., research assistant, teaching assistant, tutor, etc.). It would be good to attach an updated and relevant resume, and just thank them for their time and consideration.
I hope that helps. Good luck!