An argument is a textual or verbal standpoint that is usually defended with factual evidence or an appeal to a person's morality.
An essay is a short piece of writing meant to express a thought or opinion (or multiple thoughts/opinions). often times it too uses evidence or an appeal to a person's morals.
A persuasive text is a text with the sole intent of convincing a person, or multiple people, to agree with an idea or opinion. It also can use evidence or an appeal to a person's ethics/morals.