If you're asking someone a question like,
Did you ate?
Did you eat?
I always have a problem with this word usage. When I want to ask someone if they ate or they didn't.
Ate is the preterite conjugation of eat and is used when conjugating without the verbs "do" or "have" (therefore, you do not use "ate" when asking a question, only when stating a fact).
Eat is the infinitive form and is used when conjugating alongside the verb "do", such as when giving a command or asking a question about whether an action took place.
Eaten is the past participle and is used when conjugating alongside the verb "have", such as when asking about the current state of the subject of the sentence or stating something that is not necessarily a fact.
As mentioned by Avery Ross, these categories can often overlap depending on the tone that you want to convey. "I ate today" and "I have eaten today" are equivalent in meaning.
As an aside, you may use ate when asking questions, but only when it is not preceded by the verb "to do" (example: "You ate last night?"). However, this is usually confusing unless said in response to something that suggests that the subject did eat last night, such as an empty takeout container in the living room. When asking if the subject ate after talking about something unrelated, you should use eat (example: "Did you eat last night?").