There are several ring species, but the most famous example is the herring gull. In Britain, these are white. They breed with the herring gulls of eastern America, which are also white. American herring gulls breed with those of Alaska, and Alaskan ones breed with those of Siberia. But as you go to Alaska and Siberia, you find that herring gulls are getting smaller, and picking up some black markings. And when you get all the way back to Britain, they have become Lesser Black-Backed Gulls.
So, the situation is that there is a big circle around the world. As you travel this circle, you find a series of gull populations, each of which interbreeds with the populations to each side. But in Britain, the two ends of the circle are two different species of bird. The two ends do not interbreed: they think that they are two different species.
There are other examples. There is a bird species which rings the Tibetan plateau. At one point north of the plateau, the two ends of the ring do not interbreed.
The most-studied American example is a salamander:
Moritz C., C.J. Schneider, and D.B. Wake. 1992. Evolutionary relationships within the Ensatina eschscholtzii complex confirm the ring species interpretation. Syst. Biol. 41(3):273-291.There are seven recognized subspecies of this salamander, arranged around the central valley of California. At the southern end, the coastal and inland populations do not interbreed.
The explanation for all this is straightforward. Two species are the same if there is "significant" gene flow between them. But there is no sharp dividing line between "significant" and "insignificant".