It depends on the magazine. Many publications have submission guidelines, and you might check those for what anything required in the cover letter. I found an interesting variety of requests with a few minutes of googling.
Asimov's Science-Fiction is clear on what they want:
Your cover letter should contain the length of your story, your publishing history and any other relevant information (e.g., if you send us a story about a medical disaster and you happen to be an emergency room nurse, mention that.).
The Briar Cliff Review is a little vaguer:
Please send a cover letter and short biographical note with all submissions.
McSweeny's is nearly as vague, but more interesting:
Please keep [your cover letter] brief, though we do like to hear from people who read and like the magazine. We're not concerned about writing degrees or past publications, though, so don't be daunted if you don't have an MFA or much in the way of previously published work.
In short, make certain to read the submission guidelines. (Online and print requirements may differ, even within the same publication.) A manuscript in the wrong format is not going to be read quickly, if at all. I don't know if the tone of the cover letter matters or not, although if I were submitting to McSweeny's I'd spend more time on that letter.