For most of the 20th century, the two countries known today as the Czech and Slovak Republics existed as one country, Czechoslovakia. After the end of the communist era in 1989, desire for self determination resulted in a division of the country in two. On January 1, 1993 Czechoslovakia officially ceased to exist and the Czech and Slovak Republics were born. This is sometimes called the "Velvet Divorce" since the entire process occurred in a peaceful and orderly manner.
Though each country has its own history and language they share many similarities. The languages for example, though slightly different, are similar enough to allow easy communication. Historically there has always been a lot of flow of people and ideas. Even today despite the countries being independent of each other, there continues to be a special closeness between the two nations.
Learn more about the Slovak Republic: http://en.wikipedia.org/wiki/slovak_republic