When does a past participle have to agree with other parts of the sentence?

Past participles are interesting because they are used in several different ways. The most important of these is in the passé composé, but you can also find them behaving as adjectives.

Le garçon est perdu.
La plume est perdue. 

Here you can see the past participle of perdre behaving like an adjective, which means it has to agree in gender and number with the thing is is describing. Of course, you can also find past participles as part of the passé composé, and here there are _ types of sentences when there has been agreement. 

1. When the auxiliary verb is être and not avoir. E.g Le garçon est allé. La fille est allée. The past participle has to agree like it was an adjective. 

2. When there is a preceding direct object. E.g. Je l'ai trouvée. Let's imagine that the l' here refers to 'la plume' and therefore is feminine. We have to make the past participle agree, and so we add an e in this case.

3. When it appears as part of a relative clause. This is perhaps the trickiest of the lot, but it's impressive to master. This is when the relative clause (using the passé composé) refers to an object. It's easiest to look at an example. E.g. C'est la plume que j'ai trouvée. It's like the object before the 'que' is treated as a preceding direct object.