Putting on airs as a term refers to someone adapting an elitist and arrogant tone. By virtue of its definition it's almost universally used as a disparaging remark.
The term derives from the French word air, itself a derivation of the Latin word ager, essentially meaning "field" (other derivations: acre and agriculture). The French word then took on a bit of the more common meaning of the word air and came to mean something akin to "appearance" or, more colloquially, the "vibe" of something. (Consider the "air of mystery" so common to Sherlock Holmes and Agatha Christie.)
At the same time, the roots of the common air involve "lifting" and "rising", and in this sense, putting on airs conjures up a deliberate attempt to raise one's station, albeit artificially - one is merely "pretending" to be something that one is not. This charge is tinged with the undertone of hypocrisy - that a person putting on airs is often doing so at the expense of their own true beliefs and personality.
In Richard Lederer's book Crazy English, he noted that when a single word has multiple meanings but one meaning is majoritarian, language will eventually evolve to exclude the lesser interpretations - Darwinian linguistics, if you will, and "putting on airs" falls in this category. The phrase, while not archaic, has faded from the vernacular, replaced by snappier and more allusive putdowns ("snotty", "holier than thou", and the proverbial high horse among them) that don't make tortuous use of the word "air". But you can still use it to put someone in their place. Just don't put on any airs yourself when you do.