Learn Russian: Мы написали ему письмо.

ему is the Dative case form of он (he). We use the Dative case to show the receiver of an action (the indirect object). Just like in English when you say We wrote him a letter, Russian uses the Dative case to show who is receiving the letter, without needing an extra word for to. его is the Accusative case, which is used for direct objects.

Why does the verb написали end in -ли?

In the past tense, Russian verbs do not change based on who is speaking (I, you, he), but rather by gender and number. Because the subject here is мы (we) which is plural, the verb takes the plural past tense ending -ли.

Is the word письмо in the Nominative case (the dictionary form) here?

It looks like it, but it is actually in the Accusative case because it is the direct object, meaning it is the thing being written. For neuter words like письмо, the Accusative case is spelled exactly the same as the Nominative dictionary form.

Could I change the word order and say Мы написали письмо ему?

Yes, Russian word order is very flexible. However, the order in our example is the most standard, neutral way to say this. Moving ему to the very end places special emphasis on it, making it sound more like We wrote a letter to HIM, and not to someone else.