Learn Russian: У секретаря нет документа.

This is because of the Genitive of negation. When you use нет to say that someone doesn't have something, the missing object must be in the Genitive case. Since документ is a masculine noun ending in a consonant, we add -а.

Why is the secretary also in the Genitive case (секретаря)?

Russian doesn't use a verb for "to have". Instead, possession is shown using the preposition у (meaning "by" or "at") followed by the owner in the Genitive case. So, у секретаря literally means "at the secretary". Both the owner and the missing object happen to use the Genitive case in this sentence, but for completely different reasons!

If секретарь is masculine, why does it end in -я here while документа ends in -а?

It depends on the final letter of their basic dictionary forms. Документ ends in a hard consonant, so it takes the standard -а ending in the Genitive case. Секретарь ends in a soft sign (-ь). To keep the ending soft, we drop the -ь and add -я.

How do we know it means "the secretary" but "a document"?

Russian does not have articles like "a" or "the". We figure it out based on context and word order. The person at the beginning of a Russian sentence is usually already known ("the secretary"), while new information is placed at the end ("a document"). Depending on the situation, translating it as "the document" would also be completely correct.