In hanja, the name looks like this:

In Mandarin Chinese, these characters are read Dahan Minguo, and in Japanese, they are read Daikan Minkoku.

Taken very literally, the four-character name means "State of the Great Han People." It is often contracted to Hanguk, "Han State," in less formal situations (and, likewise, you have Hanguo in Mandarin and Kankoku in Japanese).

Now, here's some kimchi for thought. In English, we have only one word for Korea, but there are actually several Chinese characters that can be used to refer to Korea. South Koreans use the Han character to refer to their country. North Koreans use the name 朝鮮 Joseon, which is the most common way to refer to Korea as a whole in both China (Zhaoxian) and Japan (Chôsen). In Japan, there's been a considerable amount of debate over whether to call the Korean language Kankokugo, as the South would, or Chosengo, as the North would: in fact, the parastatal broadcaster NHK usually calls the language Koriago to escape the debate entirely.

Another interesting point is that the South Korean word for "republic," minguk, uses the same characters as the Taiwanese word, míngúo. However, North Korea's word for "republic" is gonghwaguk, literally "state of cooperative peace," which uses the same characters as the Japanese term, kyôwakoku, and the PRC term, gonghuoguo.

And you wonder why there are so many damn soldiers in the DMZ.