When should there be verb agreement?

In French, the past participles in compound tenses and moods often need to agree with other aspects of the sentence. You need to know both how to make them agree, and also when they should agree. Verb agreement is quite similar to adjective agreement, in that for feminine endings, you add an e, for plural endings you add an s, and for feminine plural endings you add es. Let's discuss when verbs should and should not agree.
Verbs that take être in the past tense must always agree with the subject. For example: La femme est arrivée au restaurant. Ils sont allés au cinéma.
Verbs that take avoir in the past tense usually don't agree with the subject, unless the direct object precedes the verb. For example: La pomme que j'ai mangée. Les lettres que tu as lues hier.
Pronominal verbs (those that use reflexive pronouns) sometimes must agree, and other times must not. In theory, as pronominal verbs take être in the past tense, they must agree. However, this is only the case if the reflexive pronoun is a direct object; if it's an indirect object, there is no agreement. For example: Elle s'est lavée - there is agreement because the reflexive pronoun refers to the person that washed themselves. Elle s'est lavé les cheveux - there is no agreement because the reflexive pronoun refers to the hair that has been washed.
Pronominal verbs that, when used non-pronominally with a noun, require a preposition (à, de etc.), there is no agreement as the preposition and indirect object are replaced by a reflexive pronoun, so the pronoun is also indirect. For example: Ils se sont parlé. - parler à quelqu'un, indirect so no agreement. La femme s'est acheté une nouvelle voiture. - acheter pour quelqu'un, indirect so no agreement.