The passé composé is the most common past tense in spoken French, used for completed actions: j'ai mangé (I ate), elle est partie (she left). It's a compound tense, built from an auxiliary verb in the present (avoir or être) plus the main verb's past participle.
Most verbs use avoir as their auxiliary — j'ai parlé, tu as fini, il a vendu. A small, closed set of about seventeen verbs (mostly verbs of motion or state — aller, venir, partir, arriver, entrer, sortir, monter, descendre, naître, mourir, rester, retourner, tomber, and a few others) use être instead, and when they do, the past participle agrees with the subject like an adjective: elle est partie, ils sont arrivés.