What does it mean that there was "no Republican primary"? That there was no way for people in those counties to vote for e.g. Rick Perry in the primary? Surely not. Is it local offices?
jimrtex has a good explanation in Miles's link. But yes - basically, there is no way for people in the dark red counties with asterisks to have voted in the Republican Primary, for any office at any level, even if they wanted to, because the GOP had no county chair in those counties. (Though if you can't even manage to find someone willing to be the chair, which in a small county is a pretty nominal post with little actual obligation, there probably aren't many people who want to vote in that primary to begin with.) And conversely, the people in the dark blue counties with asterisks could not vote in the Democratic primary because the Democrats had no chairs in those counties.
This isn't a great system, particularly for rural counties that are so short on people. In 2012, there was some controversy over the Republican primaries in a couple of West Texas counties that are heavily Republican not taking place because the county chair unexpectedly moved out of the county and a replacement couldn't be found in time. In another county, the party chair moved away between the primary and the runoff, meaning citizens there got to vote in the primary but not the runoff.