The Republicans want to repeal the 14th amendment to the Constitution, the one that promises equal protection under the law and the one that confers citizenship on anyone born in the country, regardless of the status of the child's parents. Of course, the Republicans have only themselves to blame for the existence of the 14th amendment. It was an initiative of their party after the Civil War. It was meant to protect the status of newly freed slaves and their children. It was also used to rehabilitate those who had rebelled against the United States. Confederate soldiers and sympathizers were able to take an oath of allegiance to the Constitution of the United States and they were essentially "reinstated" as citizens with all the protections that that conferred. Yes, this was an initiative of Abraham Lincoln and the Republicans of the 1860s to heal the sundered Union. Today's Republicans would evict him from their party.
Not only do the Republicans want to repeal the 14th amendment, it often seems they want to repeal all of them. (Except the second, of course. They looooove that one!)
In fact, when you get right down to it what they would really like to do is abolish the entire Constitution and substitute the Bible. Some of them will even come right out and say that, especially if they are at a tea party gathering. I don't think they've really thought that through, though, for the Old Testament punishment for some of the Republicans' favorite sins is rather harsh. They might find their ranks suddenly reduced by stonings if they introduced the strict Jehovian rules.
Of course, all of this is just politics. They are just trying to appeal to and stir up their old white right-wing base and it might work well enough in the short term. But in the long term, those old white right-wingers are going to shuffle off their mortal coil and the Republicans are going to be left to deal with a younger, darker, and more liberal electorate. What will they do then? Tack to the left? For people who will say and do anything to get elected, that won't be difficult at all, for they have no true moral center. They really are repellent.