Learn Russian: Он быстро написал письмо.

The verb написал is perfective. We use the perfective aspect here because the action was completed successfully—he finished writing the letter. If we used the imperfective verb писал, it would mean he was writing the letter (focusing on the process) or that he wrote letters regularly.

How do I know написал is in the past tense?

In Russian, past tense verbs almost always end in -л. The exact ending changes based on the gender of the person doing the action. Because он (he) is masculine, the verb takes the basic masculine past tense ending -л. If a woman had written it (она), it would be написала.

Why doesn't the word письмо change its ending here? Isn't it the direct object?

Yes, it is the direct object, which means it is in the accusative case. However, письмо is a neuter noun (it ends in -о). Neuter nouns do not change their endings in the accusative case; they look exactly the same as they do in their basic dictionary form.

Does the adverb быстро always have to go before the verb?

Usually, yes. In Russian, adverbs that describe how an action is done (like быстро, meaning "quickly") typically go right before the verb they describe. You might see the adverb placed after the verb occasionally for special dramatic emphasis, but putting it before the verb is the most natural, standard word order.