Here is a letter that I made up as an example. Comments to each individual part are in the right hand-side column. See also general comments below.
Text of letter
Dear Dr. Miller,
For the US, use Dr., not Prof., to address someone.
I am a second-year student of Biochemistry and Cell Biology at Jacobs University Bremen, Germany.
Introduce yourself first.
In order to gain practical experience in a research environment and to explore graduate school options, I wish to perform a 10-week internship in a research laboratory, and I would like to ask whether there might be a place available in your laboratory from June to August, 20xx.
Tell the addressee what you want from them, and why.
During my studies at Jacobs, I have become interested in studying the molecular mechanisms of the immune response. I find it especially interesting how one might manipulate the system in order to induce specific immunity against viruses or other pathogens. The molecular basis of this is antigen presentation through MHC class I molecules, and I would like participate in research on them because it combines techniques across the whole spectrum of the life sciences, from molecular dynamics to cell biology.
The next question the addressee will ask is "Is he or she really motivated to work with me, i.e. applying specifically to my lab or am I just one of many people who get a nonspecific letter?"
I came across your laboratory when reading the literature on MHC class I molecules, and especially the work in your group described in the Journal of Immunology, 2004 ("The role of class I molecules in ankylosing spondylitis") and in Nature, 2005 ("Class I haplotype determines clinical outcome of ankylosing spondylitis") has motivated me to write to you.
Here, you should demonstrate that you know some of the work from the lab, ideally by mentioning the titles of a couple of interesting papers (make sure you have actually seen them).
I would be very excited to participate in work on a similar subject.
Do not suggest an actual research project - as an intern or beginning graduate student, you have no control over that.
Regarding my training, I am convinced that the intense teaching approach at Jacobs University has equipped me with solid fundamental knowledge of biochemistry and cell biology. Since Jacobs puts its students on the track to a career in research, this is complemented by laboratory courses from the first semester on. I have also worked as a research assistant in Prof. xxx's lab for xxx months to gain some additional experience.
The next question the addressee will ask is "Is that person qualified to work in my lab or will they just break everything?" So you must convince them that you are well-trained. Of course, only write what actually applies to you.
If I am given the chance to work in your group, I am certain that due to my previous training and my enthusiasm for the subject, I will be able to make a meaningful contribution to your research effort.
This part is important. What is your host going to get from your time with them?
I have attached my CV, which contains the details of my theoretical and practical training, my current university transcript, and the names and addresses of three referees. If you wish, I can ask them to send their letters directly to you.
Normally, for an internship, the prospective host should contact the referees and ask for the letters. (This also helps minimize the impact on Jacobs faculty who have to write many letters for grad school already.)
I look forward to hearing from you.
General comments on style:
© Sebastian Springer 2005-19. Please contact me with suggestions for improvements.