In my endless search for internships, I spent a while last night looking at different UN sub-organizations, and read up a bit on the topic of how to get a job there. I thought others might be interested, so here are the major pathways into the UN system, as far as I can ascertain (this information is provided "as is").
Internships: Most of the UN organizations provide internships, although most of them are available to MA students only. Usually they are not remunerated, and you will have to find your own funding. Some organizations search for specific interns, whereas at others, you must fill out an online form for a database, from which interns matching certain criteria will be selected.
JPO programme: The Junior Professional Officer programme is aimed at graduates with a bachelor or master degree. It does not require extensive work experience, and is designed to familiarize the applicant with the UN system. Application is through the home government of a participating nation, and the programme is financed by the home country. The applicant is either placed in a field-mission, or less often at the central office, for a duration of two to three years. Depending on outcome, a second placement of two to three years is possible. At the end, only very exceptional participants gain final employment with the UN - the programme is also designed to train people to cooperate with the UN in international NGOs or home governments.
Competitive exams: UN personell at levels P-2 and P-3 are mainly recruited through competitive exams, which are held once a year in selected countries (depending on which nationalities are currently needed to achieve a balance in the system). Examinees must have a relevant bachelor or masters degree (students with a degree in International Development Studies are eligible to take the exam for Social Affairs. See a sample exam.). Select candidates are then invited for an interview. (Interestingly, this system reminds me quite a lot of the imperial exam system in dynastic China.)
Direct entry: Some positions at the P-3 level, and all at levels above, are filled either by external search, or internal promotion. Anyone can apply for these positions, but usually a relevant university degree plus at least six years of working experience is required.
Good luck! :~)