Learn Russian: Никто не ошибается.

In Russian, double negation is completely normal and required! When you use a negative word like никто (nobody), ничего (nothing), or никогда (never), you must also put не before the verb. A literal translation would be "Nobody does not make a mistake."

Is Никто treated as a singular or plural word?

It is treated as a singular subject, exactly like "he" (он) or "she" (она). That is why the verb ошибается uses the "he/she" ending -ет (which you can see just before the -ся at the very end).

Why does ошибается end in -ся?

The dictionary form of this verb is ошибаться (to make a mistake). The -ся at the end means it is a reflexive verb. While it doesn't strictly mean doing something "to yourself" in this case, many Russian verbs just naturally require this ending. When conjugating, you add the normal ending (like -ет here) and then tack -ся back onto the very end.