Fesses, the French equivalent of 'buttocks', is one of the most common ways to talk about one's behind in France. It's a bit colloquial, but not vulgar. If you were to write a gossip column about the celebs with the best butts out there (not saying that's a good idea), the headline would most likely contain fesses.
Fesse only refers to one butt-cheek. So unless you are aiming to be very specific, generally the correct form is the plural fesses; mes fesses – 'my buttocks'.
J'ai mal aux fesses – My butt hurts.
Frapper quelqu'un sur les fesses – To slap someone's butt.
Derrière means 'behind' as in behind something, but also THE behind (ie a person's buttocks). It's a masculine noun, so it's un derrière, and if you want to say 'falling down on one's butt, it's tomber sur le derrière.
Derrière is neither vulgar nor slang, and this is one you can safely say if you're in a formal setting (though you would probably not be talking about butts in a formal setting in France).
Postérieur is another way to say 'behind'. It's the opposite of antérieur, which means 'fore'. Postérieur is not vulgar and you can safely use it without shocking those around you.
Cul means 'ass' or 'arse' and is a slang term for one's behind. It's vulgar, but very common.
You could say: J'ai glissé et je suis tombé sur le cul – I slipped and fell on my arse.
Mon cul is also used to say something is rubbish, whereas faux cul (false bottom) means someone is a hypocrite or a phony.
Cul also has a certain blush potential for French learners when you're not saying it but people think you are. The very commonly-used word beaucoup (a lot) if not correctly pronounced sounds to French ears like beau cul (nice ass).
Generally, a pétard is a 'firecracker', as in those tiny explosive devices kids sometimes play with that make a lot of noise.
But pétard can also be a slang term for 'butt'. Péter, as you may know, can be a – very colloquial – way of saying 'to fart'. Logically, therefore, un pétard is something that makes a lot of noise, and the thing that pète (blows up). It's definitely not something you would say in front of your grandmother.
Not to be confused with the sports boules, un boule also can refer to one's 'butt'. The key here is that the butt-version is masculine, while une boule means 'a ball'.
Bouge ton boule ! – Move your ass!
Boule is a slang term mostly used by young people. There is a song called J'aime trop ton boule (I love your ass), which came out in 2007. It's actually a parody, mocking rappers such as Sean Paul's obsession with shaking butts, but it quickly became a top hit in France.
Directly translated as 'behind-train', arrière-train means 'rear' and is a slightly old-fashioned slang term for 'butt'. However this one is often used to talk about the rear end of a car or an actual train, so be aware of the context if you hear this before you assume it's a referral to someone's buttocks.
Croupe means 'rump' and usually refers to the behind of a horse, but it can also be slang for the human rump. It's similar to derrière and postérieur, but it's more of a slang expression when used on humans.
There's also croupion, which refers to the 'tail' of a bird, but also can be used as a colloquial slang expression for 'butt'. J'ai mal au croupion – My butt is hurting.
Popotin is a quirky way to say 'butt'. It's not vulgar, but a common, colloquial slang way of saying postérieur, which originated sometime in the 20th century. Remue-toi le popotin is a nicer way of saying bouges ton cul – move your ass.
Une miche de pain is a loaf of bread. The loaf in question is round and quite big, just like some human posteriors. Les miches sometimes therefore is used to talk about the les fesses. Like most of these expression miche is a slang term, but not a vulgar one.
Lastly, there's lune, 'moon', which is pretty self-explanatory. Voir la lune en plein jour is an old and common French expression that directly translates as 'to see the moon during daytime', but actually means 'to see someone's naked backside'.
You might be familiar with the English expression 'mooning' as in 'flashing one's butt'. In French, you would say montrer sa lune – showing one's butt.