Agree in writing what your job responsibilities will be. Be sure that this agreement represents what you wish to get from the internship as well.
Evaluate your experience to make sure your are getting from it what you expected. Meet regularly with your supervisor to discuss whether you are meeting your goals and their goals, get feedback on your performance, and to clarify assignments and tasks before heading in the wrong direction.
Be a dependable employee! For example, don't forget to call if you're sick or going to arrive late. This lets your supervisor know your job is important to you.
Plan to work hard and do more than is expected. Avoid "it's not my job" thinking. Take the initiative and ask for additional assignments once you've completed yours.
Learn the "corporate culture" of your organization and be willing to adapt to it.
Develop and utilize strong interpersonal communication skills, including both oral and written communication. Learn to listen carefully to be sure you fully understand instructions or requests.
Demonstrate patience and sensitivity with others, avoiding public confrontations or complaining. Learn the art of tact when working on a team which includes being receptive to other's input and knowing when to compromise.
With written work, be sure you've proofread it carefully for content, spelling and grammar before submitting it.
Another aspect of good communication skills is developing relationships with your coworkers. Get to know and be known by others in your work environment...your professional growth depends on it.
Leave your internship with a good recommendation from your supervisor and stay in touch - networking is the heart of a good job search!
Finally, the key to a successful transition into your new internship is to take the initiative and do the best you can.