Stressed (or disjunctive) pronouns — moi, toi, lui, elle, nous, vous, eux, elles — stand in for a person when a subject pronoun can't be used grammatically: after a preposition (avec moi, with me; pour toi, for you), after c'est (c'est lui, it's him), or alone as a one-word answer (qui a fait ça? — moi, who did that? — me).
They're also used for emphasis, stacked alongside the regular subject pronoun: moi, je pense que... (as for me, I think that...). This kind of doubling has no real English equivalent — it's simply how French adds stress to "I" or "you" in casual speech.