Learn French: Je vois le videur de la discothèque.

It literally translates to "the emptier," coming from the verb vider (to empty). It makes perfect sense when you think of their job as "emptying" the club of troublemakers! Another term you might hear in France is le physionomiste (or le physio), but that specifically refers to the door person who decides who gets past the velvet rope based on their appearance, rather than the security staff breaking up fights.

Is discothèque the most common way to say "nightclub"?

It is widely used and understood, but in everyday conversational French, you are actually much more likely to hear une boîte de nuit or simply une boîte. You could just as easily say Je vois le videur de la boîte de nuit.

I sometimes still mix up de, du, and de la. Why is it de la discothèque here instead of just de?

This is a great chance for a quick refresher! You are saying "of the nightclub" to specify exactly whose bouncer it is. Because discothèque is a specific feminine noun, it requires the definite article la. The preposition de (of) and la (the) remain separate as de la. If you were talking about a masculine location, like le club or le bar, it would strictly contract into du (le videur du club).