The Konik horse has an interesting history. It is the result of breeding programs which were attempting to resurrect the extinct Tarpan horse, one of the original wild horses of Europe. The last Tarpans died in captivity in the late 1800s, but interested parties noticed some horses in rural areas of Poland which resembled the Tarpan, though they were probably mixed with domestic bred horses. These were gathered for a breeding program to try to recreate the Tarpan, with one result being the Konik (which means "little horse" in Polish). According to this article, the Nazis, fascinated with nature, ancient history and genetics confiscated some of the herds during WWII, taking them back to Germany. These herds did not survive the war, but enough Koniks remained in Poland to resurrect the program after the war ended, and today there are Konik herds in various parts of Europe.
There are more beautiful photos of Koniks on Astrid van Wesenbeeck's photostream on flickr, which is the source of the photo above.The horses in her photos live in a nature reserve in the Netherlands.
There have been further attempts even in the United States. The North American Tarpan Association supports breeding and registering of Heck horses, although it may not be active anymore, as the last blog posts are from 2006.
Another breeding program in the US is run by the Stroebels family in Oregon, about which more in this news article.