Passé composé: how does the agreement of the past participle work?

The passé composé (present perfect) is a frequently used tense to refer to a finished action, event, etc. in the near past. In spoken French, it often replaces the passé simple (past historic). 

It is a compound verb form made up of a conjugated auxiliary (être or avoir) and the past participle of the verb.

Example: Hier, je suis allé à l'école. [first person singular of "être" + past participle of "aller"]. 

The rules of the past participle's agreement vary. The basic ones are as follows: 

With être, the past participle agrees with the subject of the verb.

Examples:

Elles sont parties à Paris. 

Pierre est tombé à vélo. 

With avoir, the past participle agrees with the direct object only if it comes before the verb.  

Examples:

Tu as vu la nouvelle moto de Romain? Il l'a achetée hier. ["la nouvelle moto de Romain" is the direct object; in the first sentence, it is after the verb, no agreement; in the second sentence, the personal pronoun " l' " is the direct object, replacing "la nouvelle moto de Romain"; the past participle "acheté" therefore agrees with it.]

However, the rules change when the verb is reflexive (always used with être). 

When the subject of the verb is also the subject of the action, the past participle agrees with the subject.

Example:

Ils se sont lavés. = They washed themselves.

[Who/what is being washed? --> "Ils". The subject is therefore the recipient of the action, there is agreement.]

When there is a direct object that is the recipient of the action, then the agreement rules are the same as with avoir: the past participle agrees with the direct object if it is placed before the verb, and doesn't agree if it is placed after it.

Example:

Ils se sont lavé les mains. = They washed their hands.

[Who/what is being washed? --> "les mains". "Les mains" is the direct object and is placed after the verb, no agreement.]

Les mains qu'ils se sont lavées. = The hands that they washed.

[Who/what is being washed? --> "les mains". It is the direct object and is placed before the verb, so the past participle agrees with it.]