As a UX Design Intern at Microsoft, you are part of a design team working on a product. You are treated much like a full time employee, except your project has less constraints. My days were usually filled with one or two meetings (with designers or other people related to my project), an occasional intern event (speakers, volunteering, mixers, games, tours, etc.), coffee-chats/meet-ups with full-time employees in other departments, and work time at my desk. During work time I am usually writing emails, doing research, conducting interviews, sketching prototypes, making slide decks and creating mockups.
It varies, depending on where you are in your project! On days when I was conducting user studies and talking to many people, I found myself traveling back and forth across the company campus with my computer, graph paper, and a pen. Other days, when I was focusing more on wireframing, I'd be sitting at my desk and take intermittent breaks to hang out with the other interns / my coworkers. The nice thing is that the Microsoft Intern program coordinates a lot of different events for its summer interns; I found that as a UX intern, I actually had less time to attend those events than my software engineering and program manager intern friends, but what I was doing was just as fun.