I agree with Diego in certain things, however here are my 2 cents:
A chiste is merely a joke, generally intended to make somebody laugh rather than make fun of someone, but both uses can be covered by this term.
Les conté un chiste pero nadie lo encontró divertido.
I told them a joke, however nobody found it funny
Hice un chiste inocente acerca de los dientes torcidos esperando que mejorara el ánimo de Lauren, pero lo único que logré fue que se enojara.
I made an innocent joke about crooked teeth hoping to brighten up Lauren's mood, but all I accomplished was to make her angry.
In the same way, chanza (though not commonly used in everyday speech—— except in Mexico, I think, where they have adopted it as synonym of oportunidad thanks to the similarity with the English noun chance) can be understood as a chiste or as a witty remark; which leads me to think that the closest word in English for it depending on the context would be witticism. I'd dare to say you should opt for chiste instead of chanza, there's a common saying that uses both in certain countries and only one in others.
Entre chiste y chanza la verdad se asoma.
Many a true word is spoken in jest.
Aditionally, broma can also be replaced in the aforementioned saying.
Entre broma y broma la verdad se asoma.
However, it's also used in contexts where you literally play a prank on someone.
Le jugué/hice una broma para su cumpleaños y me dijo que había sido muy original.
I played a prank on him for his birthday and he told me that it had been a very original one.
Like Diego said, broma in this context is not meant to make fun of somebody but rather to make somebody laugh despite the circumstances. Then again, not everyone shares the same humour and somebody might not take it the right way.
This one is more specific. It has a slight negative connotation, since whenever you use it, you give away the impression that it is indeed your intention to make fun of someone/ridicule/mock someone.
Nos burlamos de sus gafas que hacían que su nariz se viera más grande.
We made fun/mocked him for his glasses since they made his nose look bigger.
Finally, as for mofa and guasa, I can say I've rarely ever heard any of them. Perhaps I've heard mofa in literature,but I can say this is the first time I see guasa. Though the DRAE (Diccionario de la Real Academia) defines mofa similar to burla, that is, it might have a strong negative connotation in its usage.