The word "complement" means something that completes (something else). For instance, I could say that ketchup perfectly complements my french fries. (Or my steak, if I'm a boor like the president LOL)
In grammar, the word has a very similar meaning.
A subject complement is an adjective or noun that appears in the predicate but refers back to the subject:
The king is a tyrant. (The noun Tyrant refers back to king and is its complement)
The king is sick. (The adjective Sick refers back to king and is its complement.)
Note that in the sentence "The king ruled his empire," Empire is NOT a complement, as it does not mean King. Instead it's a direct object.
Jimmy's teacher called him a troublemaker. (Troublemaker is a complement of the pronoun Him. Him is an object, so Troublemaker is an object complement.) If you find this one confusing, go back and refresh on what a direct (or other) object is.
The accusation made me angry. Subject: accusation. Verb: made. Direct object: me. Object complement: angry (since Angry refers back to the noun Me.).