Learn German: Ich brauche ätherisches Öl, wenn ich müde bin.

Here we have a neuter noun (Öl) in the accusative case, since it is the direct object of the verb brauchen. Because there is no article like das or ein in front of it, the adjective has to do the heavy lifting to show us the gender and case. It takes the strong ending -es, which echoes the -s in the definite article das.

If ätherisches gets a special ending, why doesn't müde get one too?

Adjectives only change their endings when they are placed directly before a noun to describe it (like ätherisches Öl). When an adjective comes after a linking verb like sein (to be) to describe the subject, it stays exactly in its basic dictionary form. This is called a predicate adjective.

Why is the verb bin at the very end of the sentence?

The word wenn (when/if) is a subordinating conjunction. It creates a dependent clause, which kicks the conjugated verb (bin) all the way to the end of that clause. The first part of the sentence (Ich brauche ätherisches Öl) is a main clause, so its verb (brauche) stays in the normal second position.

Could I use wann instead of wenn here?