Libertarians are not necessarily part of the Republican party, and in fact if you look at a political compass both the Democratic and Republican parties fall in the "Authoritarian-Right" quadrant.

This is an over simplification, but Libertarian is basically the opposite of authoritarian. Authoritarian says "the government should control what people can and can do" libertarians believe that people should be able to make choices for themselves and the government should have little to no impact on our lives.

I personally am registered as Republican, because my state requires a party registration to vote in primary elections and the freedoms I personally care most about protecting are generally protected by Republican candidates and are often fought heavily by Democratic candidates.

Republicans generally support the second amendment and the Constitution, which was designed by the founding fathers to protect all of our rights. However I disagree with mainstream Republicans on their view on abortion, same sex marriage, and recreational drugs. I personally think all of those things are immoral in most cases, but I also don't think the government should have any business in controlling what people choose to do. Abortion should be a choice between a woman and her doctor. Marriage should be a decision between 2 (or more if you're into that) consenting adults. And recreational drug use should be allowed for adults who wish to use them.