Research and teaching statements

UIUC's Career Services Office has a lot of stuff about this topic.

What's a research statement?

This is a brief summary of your work, which:
1) orients the reader to your specific interest within a broader discipline
2) describes how your research is related to others' work (though avoid discussing your dissertation adviser's work exclusively - be sure you represent yourself as the independent researcher that you are)

The statement should:
1) indicate why your research is important
2) identify who might care about it
3) provide your top 2 or 3 research questions
4) present the plans you have for addressing them in the coming years (make sure it sounds more involved than continuing to replicate findings from your dissertation)
state any funding sources you have in mind

What's a teaching statement?

This is a brief description of:
1) what you do in the classroom and why you do it, (do not simply repeat your course offerings)
2) your method(s) of teaching
3) your beliefs or justifications about why the method(s) are valuable
4) your specific goals as a teacher and the corresponding approaches you use to meet these goals
5) any technology you use in the classroom
This statement should demonstrate that you have been thoughtful about your teaching process.