Doing away electives abroad is probably much easier during medical school than residency, although some residencies have good contacts or formal affiliations with schools abroad for the opportunity.
Since your question is rather broad based, I suggest you narrow it a bit (ie, what country, what field, residency vs medical school) to increase your chances of getting a response. Most US med students and residents DO NOT go abroad at any time for their education so you may find more information and knowledge in the IMG forums here.
I traveled to Costa Rica with Amerispan during the summer after my first year of med school. It was a wonderful experience.
Pretty soon, I'll be leaving for Ecuador with Child Family Health International. I'll be doing a community medicine rotation, but they have rotations available in tropical dz, surgery, and something else. The organization also has rotations in Mexico, South Africa, and India. They're all pricey though--the Mexico one is the cheapest, and teh India one is very expensive .
You should check the AMSA website. A couple years ago, I contacted people in Africa, Australia, Thailand, Nepal, S. America, and many others. I heard back from most of them too.
And it IS possible to do electives abroad during residency, depending on which specialty you apply for. I plan on doing this in residency, and only ranked programs that would allow their residents to travel abroad.
Hope that helps. Let me know if you have other q's.