When you are given a task, make sure you complete it on time and that you are making every effort for your work product to add-value. One thing in particular would be to find inefficiencies in any day-to-day process, learn how to make that process or project more efficient, develop your comprehensive solution and present it to your manage. Every work product you submit should be as if you are submitting a paper or project to a professor, but he says he will only accept a work product that would be graded as an "A". Pay keen attention to instructions and if you have questions after, right them down and ask them all at once. Avoid extended periods of back and forth with questions trying to figure out the best way to handle a task or project. Ultimately every intern (including myself) makes mistakes, and that is expected. The fewer you make, less often you make them and most importantly, as long as they aren't the same mistakes you've already been corrected on and you've shown the ability to adapt, be resourceful and provide solutions, your manager will want to keep you around. It also helps to develop the personal relationship, develop a rapport. To some people that part is the most important. Hope this helps.
If you already have an internship I think you are well on your way to receiving an offer in my opinion. To ensure you parlay that internship into a full time position I would suggest that you are not afraid to ask questions of the people you are working with, try to understand why they are doing what they are doing. Ultimately if you can make the person you are working with’s life easier then you have succeeded in making the most of your internship. Try to learn and be engaged, if you do that then you have done all you can do try and make it work.