Familiarize yourself with the magazine or company.
Okay, this sounds obvious, but you'd be surprised. Make sure you know what the magazine or company you're interning for is all about. If it's a mag, read the print (or online version) as much as you can and know it from front to back. Understand who the readers are, what kinds of topics it covers, and which brands or trends appeal most to its audience. If you applied for this internship simply to put the name on your resume, you're already making a mistake.
You have to actually care about the magazine and that which it covers. It'd be even better if you're a fan. You'll be able to do a better job and would be more useful if you knew exactly what types of things the publication, or company, does.