The first tip for a convincing cover letter is to know your target audience. Spend some time finding out about the organization where you want to intern and the industry they are involved in. Read the position description and understand what kinds of skills the organization needs for that position.

The second tip is to know yourself, and help the organization know you better through your cover letter. Your resume gives an outline of your past experience. The cover letter is your chance to tie it all together. You can give more information about what you have learned and applied in your coursework or prior positions. It is also your chance to talk about what motivates you to apply for this internship and what inspires you about the organization.

A cover letter for an internship through IE3 Global Internships is somewhat different because it will be passed along as part of a packet of information with an introduction by the Regional Director, so you don't have to focus as much on how you found out about the opportunity or what position you are applying for.

Additional Tips:
Visit the career center on your campus to see samples and get help writing a strong cover letter. This is a skill that will help you for the rest of your career. Learn to do it right the first time!

It's not all about you. Don't focus only on what you would get out of the internship. Emphasize what you offer the organization. It is fine to mention what you hope to learn as a way to demonstrate your understanding of the organization, but don't say you want this position because it will help your future career. That is overstating the obvious.

Keep it to one or two pages. The first paragraph should include general information about the opportunity you are seeking. Introduce your interest in the organization. The second and third (and possibly fourth) paragraphs should demonstrate your understanding of the organization, talk about your skills and expeirence and demonstrate what you have to offer them as an intern. In the closing, politely thank them for considering your application.

If there is any ambiguity in your resume (e.g. and unexplained gap in schooling or work) use the cover letter as a way to explain what you were doing, especially if it relates to your proposed internship application (e.g. travel, volunteer work, etc.).

Online resrouces with Cover Letter suggestions include Monster and About.com.