Typically funding comes in the following forms:

1. Fellowship/Scholarship: gives you flexibility in terms of research projects.
2. Research Assistantship (RA): aka your professor is paying for you.
3. Teaching Assistantship (TA): typically involve about 10-20 hours/week between teaching a recitation session, grading homeworks and office hours.  

For incoming grad student, if you are a domestic student and don't have external fellowship already, definitely consider applying for one now. There are plenty of such opportunity available for current graduate students, and it really helps you in the sense that professors have limited funded projects and if you have your own funding then you might even able to create new projects.

Resources for locating fellowship opportunities:

• NSE fellowship list: http://web.mit.edu/nse/education/scholarships/index.html
• ODGE fellowship list: http://odge.mit.edu/finances/fellowships/odgefellowships/

Fellowship for international students are tricker. Still check out the above lists in case anything new pop up, but as far as I know it essentially comes down to the fellowship programs provided by your own country/government/organization.

Every once in a while you would get an email from Clare about new fellowship opportunity, so definitely watch out for those. For instance, in Sept 2014 National Research Council annouced fellowship opportunities, some may be open to foreign nationals (see link here).