The terms third-world and first-world should be avoided in formal communication, except when their original meanings are intended. In some contexts, they may be seen as offensive code words.
When coined in the 1950s, First World denoted the Western democracies and countries aligned with them, Second World denoted the Soviet Union and its allies, and Third World referred to all nations not aligned with either side. But today these terms’ original senses are all but obsolete, and third-world is often used euphemistically to refer to poor, culturally unfamiliar places. This use of third-world may be interpreted as xenophobic or ethnocentric. It implies that the place being described is inferior.
Incidentally, as nouns, Third World and First World are usually capitalized with no hyphen. As adjectives, they are uncapitalized and hyphenated.
Examples
Most high-editorial-standards publications don’t use these code words, but the terms are rampant on blogs and in reader comment sections. Here are a few questionable uses of the third-world:
The site looks worse than a Third World country . . . [Galveston Daily News]
The whole trip was a complete eye opener. Being a third world country, the way of life down there is completely different. [quoted in The Sudbury Star]
These stories are playing out not in some distant third world country, but right here in the United States. [Huffington Post]
In retrospect, this has been going on for some time, mainly centred on my concern over living in the country her family live in (a terribly third-world country) . . . [Talk About Marriage]