How to apply for a research internship at the master's level?
Networking is key--check out F'x's answer to this question. Find people in your university who are willing to recommend you to the other researchers in the field, whom you would like to work with. Take advantage of the connections your faculty members have developed.

Then write a killer cover letter. aeismail notes that a compelling 'cover letter' email was the only reason he even considered an individual for a research internship. Research the PI and tailor your letter to his/her interests and current project.