What do the planets visited by the little prince represent?

Although you asked what the individual planets represent, I'll say that the planets are not so important as the people living on them. That is, the planets collectively might represent isolation and a failure to connect with others, but each individual planet's inhabitant actually represents something more. (That's probably the spirit and intent of this question.)

What the planets' inhabitants represent can be debated and interpreted somewhat differently, but I'll describe the most straightforward interpretation that most readers will have.

Each of the inhabitants of the planets that the prince visits seems to represent and expose a certain common character flaw in adults:

1. In Chapter 10, the little prince first visits the planet inhabited only by a king, who assumes that the little prince is a subject. "To them, all men are subjects," the narrator explains. This king who loves to give orders and demand obedience and yet thinks of himself as a reasonable, good man probably represents how ridiculous and petty people get when they're obsessed with being in charge.

2. Chapter 11 focuses on the prince's visit to the conceited man's planet, where its resident is only capable of hearing words that are spoken in praise toward him. This conceited man assumes that the prince is an admirer coming to adore him, and so he represents how unrealistic and irritating people get when they have an overblown opinion of themselves.

3. Chapter 12's very brief description of the tippler (an alcoholic) reveals a pitiable character who drinks to forget his shame, and who feels shame because he drinks. This tippler represents not just the futility of addiction but also the human tendency to "solve" problems by making them worse.

4. The little prince meets the businessman in Chapter 13, who's very self-important and busy in the "work" of counting the stars. The little prince notes that the businessman's thinking is just as illogical and circuitous as the that of the tippler—this businessman is focused on counting and owning things and putting them into an imaginary bank. He represents how useless it really is when people focus on amassing wealth instead of being happy.

5. Chapter 14 is about the lamplighter, who spends every moment of his life lighting the lamp, then extinguishing it, then lighting it again, and so on and so on. He does this so often because a day only lasts for about a minute on his planet, so he's lighting up the nighttime and extinguishing the light for the daytime. He's really, really exhausted but won't stop doing this meaningless work. Because of this, the lamplighter probably represents the futility and exhaustion brought on by a foolish devotion to habits or to work that is ultimately unimportant.

6. Lastly, the prince visits the geographer in Chapter 15. This puzzling character calls himself a geographer, but actually hasn't created any maps because he's just sitting around waiting for someone else to do the work of exploring, so that he can know what to put onto his maps. He's obsessed with the distinction between an explorer and a geographer, so this man probably represents the futility of clinging to rules or definitions that have no practical importance.

What you might notice as you finish up these chapters is that all of the planets' inhabitants have something in common: loneliness, and the lack of ambition to leave or change or rethink things or find companionship. All of those ambitions are held by the little prince.