The name sakartvelo (საქართველო; Georgian doesn't have capital letters) means roughly "land, or place, of the Kartvelians": the ს/s-blank-ო/o format denotes a place or land of something, and Kartveli is the Georgian name for themselves (see: ქართული, kartuli, the Georgian language; ქართველი, kartveli, the Georgian people; ქართლი, kartli, the historic central region of Georgia). Kartli dates back at least to the 5th century, while Sakartvelo is used as early as 800. That all said, I don't believe any language or people uses this name for Georgia or it's people, despite it being fairly straightforward.

Now why Georgia? There is no clear origin on the name. There are however a couple leading theories:

One of the most prominent is that it's derived from the Persian word for Georgians, gurğ (گرج), ğurğ (I can't type Persian, so this is a copy-paste job). It is through this origin that the Russian (Грузия, Gruziya) and Turkish (Gürcistan), among others, names for the country came about (note that in recent years the Georgian government has been trying to get countries to move away from this form, with Lithuania recently officially switching from Gruzija to Sakartvelas).

Another prominent theory, is that it comes through the Greek word for farmer, γεωργός (Georgós; literally "tiller of land"; again, I can't write Greek so copying this).

There is also the confusing aspect that within Georgia, the Christian St. George is a rather popular figure, so some believe that the country is simply named after him. As popular as St. George is (which is to say very; the most common male name in the country is "Giorgi"), that is not the case, only a very unusual coincidence.

If you'd like to read more in depth, I'd encourage looking into the work of Donald Rayfield. He is a now-retired English historian and translator, who worked extensively on Georgian literature and the language, including helping produce what is arguably the most comprehensive Georgian-English dictionary. His 2012 book Edge of Empires: A History of Georgia devotes the first few pages to the etymology of the country as well, and the relationship of the Georgian language itself to its neighbours (Georgian is not Indo-European, or Turkic, but in it's own family, the Kartvelian, only related to smaller regional languages). I'd also suggest Ronald Grigor Suny's The Making of the Georgian People (1988, 1994), which again opens with the discussion on the name of the country.