A colleague from my master's program at Columbia was doing her internship at UNICEF HQ in NYC. She told me that her unit was looking for someone else to help out with the work load. So, on top of my studies, a long commute to the city, and the guilty of not being a stay-at-home parent, I began a twice-a-week non-paid internship on the 4th floor of UNICEF House.
There are many requirements to become an intern at UNICEF, but the most basic one is that you have to be enrolled in a master's program. I think exceptions can be made, but I'm not sure... While you can apply directly via the website, it helps to do some research beforehand.
This is where the six degrees of separation comes in handy. It's quite certain that you know somebody who knows somebody who works at UNICEF. Find out in which Country Office and focus area you want to be interning at. This should not be an internship for the sake of internship: you need to find a match for your professional and personal interests. You're working for free, so make sure you get the most out of it!
Once you find what you're looking for, begin a conversation with your potential supervisor to agree on the terms of reference for your contract. It's much better to have this dialogue early to save any disappointments down the line and make sure both parties can gain from this experience.
Not long ago, I did exactly that when a student wrote to me expressing her desire to intern at UNICEF Mozambique. We had long skype calls and eventually narrowed down her deliverables and areas of work. She went on to do an amazing job and even published a case study based on this experience (way to go, Alejandra!).
I should add, however, that this kind of approach to internships is the exception rather than the norm. So it's up to the intern to try to get the most out of this experience, so don't settle for just an office experience - try to do something substantive (true story: I've met an intern once whose job was to come up with ideas for tweets for her boss... *sigh...).
Below is my son's take on the life of an intern...:-)
I also think it's better to have an internship at a Country Office as opposed to Headquarters or a Regional Office. You gain more 'hands-on' experience, it's a smaller work environment and you get to learn a lot more. I didn't have that luxury, so I started my UN career through the maze of a headquarters environment... More about that later!