Esperanto, the most famous and widely spoken auxiliary (or constructed) language in the world, has for over a century been popular amongst a very small but global subculture. Speakers and proponents of Esperanto say that it's simplicity and universality could allow speakers of two unrelated languages to speak to one another on equal footing; instead of a French speaker learning German to speak to a German speaker (while the German speaker can enjoy a state of linguistic superiority and speak in his native tongue), they both could learn Esperanto in very little time and then speak as equals.
Esperanto has very simple grammar; the endings of syntactic categories (parts of speech) are all uniform, so things like conjugating and making plurals are easy to do. The writing system uses a modified Latin script (familiar to much of the world), and the pronunciation is regular. Furthermore, a great amount of world literature has been translated into Esperanto and conventions are held worldwide to unite speakers.
But, honestly, very few people know about Esperanto and even fewer speak it. Schools are afraid to teach children a 'pretend' language, and since there is no notable country that has Esperanto as an official language, going abroad isn't a viable option (expect maybe using Esperanto to speak with someone you are visiting one on one). Not only that, but the claim that Esperanto is regular and easy for everyone to learn is false advertising; people used to tonal languages, non-SVO word order, people who have a different consonant/vowel inventory, and people unfamiliar with the sources for Esperanto's word base won't have a particularly easy time picking up the language. In fact, there are far more languages in the world that are dissimilar to Esperanto than those that are similar. And, as time goes on, more and more exceptions are emerging in Esperanto (so one of its main selling points is in a state of degradation).
What would happen if a language like Esperanto did catch on? Dialects. As soon as people had the ability to speak Esperanto in everyday exchanges, they would begin to make it their own and create local variations. Unless some particularly militant prescriptivists took over, we would have multiple Esperantos in a century or less; when no foreign people are around and you're just with friends, there's no reason to speak in a universal way.
The ambition is a good one: unite people and speak as equals. The reality is, though, that artificial languages are unpopular and will remain unpopular. And even if people did pick them up, they would change them so quickly there's almost no point.