Learn French: C'est le gala à la fin duquel nous avons dansé. - It is the gala at the end of which we danced.

Duquel is a contraction. The phrase for "at the end of" is à la fin de. The word "which" refers to le gala (masculine singular), so it translates to lequel. When you combine de + lequel, they contract into duquel, just like the basic words de + le contract to du.

I thought dont meant "of which". Why can't we say à la fin dont?

While dont can often replace de plus a noun, you cannot use it after compound prepositions—phrases made of multiple words that end in de, like à la fin de (at the end of), à côté de (next to), or près de (near). After these phrases, you must use duquel, de laquelle, etc.

How would this change if we were talking about a party (la soirée) instead of a gala (le gala)?

The pronoun has to match the gender and number of the noun it replaces. Because la soirée is feminine, you would use laquelle. Unlike lequel, laquelle does not contract with de, so the phrase would become à la fin de laquelle.