The correct answer to the question is option 1: Caucasus.

The Caucasus mountains are located between the Caspian Sea and the Black Sea. These mountains form a natural border between the countries of Russia to the north and Georgia and Azerbaijan to the south. The Caucasus range is approximately 1,200 kilometers long and stretches across multiple regions including the Caucasus region, consisting of the Greater Caucasus and Lesser Caucasus ranges, as well as the Southern Russia and Transcaucasia regions.

On the other hand, options 2, 3, and 4 are incorrect. The Carpathian mountains are located in Central and Eastern Europe, primarily in countries such as Poland, Ukraine, Slovakia, and Romania. The Apennine mountains are located in Italy and form the backbone of the Italian peninsula. The Elburz mountains are located in northern Iran.

Therefore, the correct answer is option 1, Caucasus, as it is the mountain range that lies between the Caspian Sea and the Black Sea.