If at all possible, get someone to introduce you to someone who works at the organization. So much happens via referrals.
Write a really clear email that explains what you are asking for and demonstrates why you are worth their time. Things to include:
- Demonstrate some knowledge of the domain (mention relevant papers you've read, events you've been to or projects you've done).
- Describe your relevant skills (e.g. technical ability, academic achievement, info about your grades).
- What career options you're considering, and how they (and/or the internship) will help you make better decisions.
Write tailored cover letters and CVs. Try to find someone to review them and give feedback (use Google Docs and put it in 'suggestion mode' so they can easily leave comments)
Read a lot of relevant stuff.
Consider starting a blog since writing online is a high impact activity. This can be great for demonstrating your thinking and getting your name out there but is more of a long term commitment.