There are 3 kinds of assistantships available to graduate students at most universities:
Teaching assistantship: This involves grading, lab monitoring, proctoring and teaching classes. Typically, every department use graduate students to help their faculty. The best way to achieve such funding would be to demonstrate excellent communication skills (both oral and written). Departments also take academic performance into account before selecting their TAs.
Research assistantship: This involves doing research as the name suggests. It would help if a student has already performed some research in a given area during undergraduate studies. This can be validated only by published papers in journals and conferences. If not, a good way would be to enroll in a professor’s class whose research interests meets yours and volunteer to work. If a student demonstrates good work ethic and productivity, the likelihood of getting this is very high.
Service/graduate assistantship: This involves students performing secretarial duties. These are available in most departments and offices located in a university. The only way to apply for these would be via career services website or by electronically/physically distributing their resumes.