Before we embark on agreement rules, let’s find out which verbs are conjugated with être (to be) rather than avoir (to have) in the passé composé. Strictly speaking, only a limited number of verbs use the auxiliary être in the passé composé. These verbs are encapsulated in the popular mnemonic device known as DR. & MRS. VANDERTRAMP: 

Devenir, Revenir, Monter, Rester, Sortir, Venir, Aller, Naître, Descendre, Entrer, Rentrer, Tomber, Retourner, Arriver, Mourir, Partir (to become, to come back, to go up, to remain, to go out, to come, to go, to be born, to go down, to enter, to go back in, to fall, to retrun, to arrive, to die, to leave)

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The basic agreement rule for these verbs conjugated with être is that they must agree in gender and number with the subject. Patricia conjugated a few of these verbs and explained how they work in her video, Le saviez-vous? - Exception dans les verbes du 1er groupe au passé composé:

Et lorsque l'on dit: "elles sont tombées",

And when we say, "they fell,"

on mettra "es" à la fin de "tombé"

we'll put "es" at the end of "tombé"

car "elles" sont des sujets féminins et pluriels.

because "elles" [they] are feminine and plural subjects.

Captions 55-57, Le saviez-vous? - Exception dans les verbes du 1er groupe au passé composé

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Knowing that the pronoun elles (they) is feminine plural makes the agreement with the past participle tombé quite straightforward, but when faced with non-gender-specific pronouns such as tu (singular you) or je (I), you need to know from context who the subject pronoun stands for. 

In the example below, we need to know who je (I) represents to establish the gender of the subject. In this case, we know from the video that the speaker is male, so the past participle doesn’t change. (A past participle is considered masculine singular by default.)

Je suis allé en Grèce pour la première fois.

I went [masculine singular] to Greece for the first time.

Caption 10, Alex Terrier - "Roundtrip" et ses inspirations

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If the speaker had been female, it would have been:

Je suis allée en Grèce pour la première fois.  

I went [feminine singular] to Greece for the first time. 

And if the speaker had been a woman talking about herself and her girlfriends, it would have been:

Nous sommes allées en Grèce pour la première fois. 

We went [feminine plural] to Greece for the first time. 

When a plural subject involves individuals of all genders, you can be faced with a dilemma. What should you do in this case? The convention is that the masculine supersedes the feminine—even though it refers to a mixture of genders, the past participle becomes masculine plural: 

Les enfants sont partis en même temps.

The kids left at the same time. 

Nowadays, however, that convention often comes across as sexist. So you'll often see past participles stylized like parti(e)s or parti·e·s to be more inclusive:

Les enfants sont parti(e)s en même temps. / Les enfants sont parti·e·s en même temps. 

The kids left at the same time. 

There's another category of être verbs that also agree in gender and number with the subject, but in a slightly different way. These verbs are called reflexive or pronominal verbs, which we will discuss in the next lesson.