Le passé composé
The most common way to talk about completed past actions in spoken French.
The rule
The passé composé is the most common way to talk about completed actions in the past in spoken French. It's formed with an auxiliary verb (avoir or être) in the present tense, plus the past participle of the main verb.
Most verbs use avoir as the auxiliary: j'ai mangé (I ate), tu as fini (you finished). A smaller group of verbs — mainly verbs of movement like aller, venir, partir, and all reflexive verbs — use être instead: je suis allé(e), elle s'est levée.
When the auxiliary is être, the past participle must agree in gender and number with the subject, which is why you'll see an extra -e or -s added in writing (elle est allée, ils sont partis).
Examples
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J'ai mangé une pomme.I ate an apple.
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Tu as fini tes devoirs.You finished your homework.
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Elle est allée au marché.She went to the market.
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Nous avons visité le musée.We visited the museum.
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Ils sont partis tôt.They left early.
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Vous avez regardé un film.You watched a movie.
Common mistakes
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Elle a allée au marché.Elle est allée au marché.
Aller takes être, not avoir, as its auxiliary.
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Il est mangé.Il a mangé.
Manger takes avoir — only a specific group of verbs (mainly movement/reflexive) use être.
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Elle est allé au marché.Elle est allée au marché.
With être, the past participle must agree with the subject — add -e for feminine subjects.