Please write a function on DropDown change. In that function pass the selected value to CAML query on which will traverse through the region list till it matches the country name.
Following is pseudo javascript code.
function onDDLChange(string countryName ) {
MyObject.regionList = MyObject.web.get_lists().getByTitle("Region");
var camlQuery = new SP.CamlQuery();
camlQuery.set_viewXml("<View><Query><Where><eq><FieldRef Name='RegionName' /><Value Type='Text'>" + countryName + "</Value></eq></Where></Query></View>");
MyObject.regionList.listItems = MyObject.regionList.getItems(camlQuery);
MyObject.context.load(MyObject.listItems, "Include(RegionName)");
MyObject.context.executeQueryAsync(Function.createDelegate(this, Success), Function.createDelegate(this, Fail));
}
in Success() you might want to just little bit of validation to see if more than one values are returned and need to get the appropriate value.