Volunteering with the Rebuild Christchurch build in New Zealand is where I feel I have made the biggest and most concrete difference (both literally and figuratively) in my time volunteering internationally.
Many parts of Christchurch were severely damaged in an earthquake in 2011, and after having organized multiple builds abroad to help other countries, NZ's Habitat for Humanity organized a build to help their own. The people and organizations we helped were mainly low income, uninsured, or otherwise in need. Seeing those people's faces each day when we showed up to our work sites was absolutely priceless.
I remember helping out at one particular house where an older couple lived--they made us tea and crumpets as a thank-you snack one day (to this day, I have yet to find a recipe that rivals that one), and we sat at talked with them while enjoying the treat. At first I was impatient to get back to work and get more done that day, but I slowly began to see (and confirmed with a fellow volunteer later) that the time we spent talking with that couple was just as valuable to them as the work we were doing repairing their home. That genuine care and gratitude has stuck with me, and has even inspired my career direction.
The skills that I learned during my time volunteering with Habitat for Humanity in New Zealand have also proved invaluable. I learned how to level and strengthen foundations, re-plaster, deconstruct and rebuild brick walls whose mortar had cracked, mix and install concrete, and much more. The Kiwi master builders had complete faith that we volunteers could learn and help with just about anything--even myself as a young-20s female with little construction knowledge at that point.
The people are really what made the experience--the people we helped, and particularly all of the volunteers, from the family who organized the builds to the people who came from around the world to help rebuild the livelihoods of people in need.