What does "Third world" mean?

What about terms like Advanced countries, less advanced and not advanced? These terms can be used to describe sociological analysis.

Yes they can, but someone would argue that economic advance is not everything and those less advanced nations are spiritually advanced or something like that.

Because that's too subjective. Although how economically advanced a nation is may be subjective (to an extent), it can be measured based on certain criteria or indicators depending on who is trying to prove what. For example, a socialist may claim an "economically advanced" nation is one that spends a lot of money on its populace (social spending), while a capitalist would say an "economically advanced" nation is one whose citizens earn the most money (GDP per capita).

More commonly in international relations we use the terms "global North" and "global South." Because a majority of LDCs are in the "South" (South America, Africa, southeast Asia, etc.) and a majority of DCs are in the "North" (Europe, North America, etc.) it is fairly commonly used without much argument (I'm not aware of any argument against this terminology). Sure, there are exceptions (Australia is considered part of the "North" while Mexico is in the "South") but the point is, in IR we need this kind of terminology to differentiate between vastly different nations in order to make assumptions... It's a social science, it's not an exact science.