Flanken-style is the way that ribs are cut, more than a cut from a specific area. The area happens to often be the Chuck area of the cow, but it can be the Rib area as well.
"English Style" Short Ribs, particularly Chuck Short Ribs, will come cut in chunks between the rib bones. "Flanken-style", by contrast, will come in a long strip between 1 and 2 inches thick (2.5 to 5 cm) cut across the ribs and across the grain, (when cut 1/4 inch / .5 cm thick, they are Korean-style.) You will see largish segments of rib bones where the saw has cut through them.
Some people feel that when cut Flanken-style, Short Ribs are more pleasant to eat. Their feeling is that being cut across the rib makes the meat easier to get at, and being cut across the grain helps to make the meat less stringy than in the standard cut. The price, however, is often close to double that of standard ribs, both for the extra work involved, and given that you often have to ask the butcher to do this cut specially for you.
This way of cutting Short Ribs is popular in Eastern European, German and Jewish cooking. One of the names for these is "Kosher-style" ribs. The ribs won't actually be Kosher, unless the meat has been processed under rabbinical supervision; the name rather reflects the popularity of this cut in the kitchens of Jewish grandmothers. Another name, "Cross-cut", is probably the most helpful name in terms of describing the cutting process.
All Short Ribs need long, slow, moist cooking methods.