Learn Russian: Она не показала это сообщение Ивану.

Because Ivan is the indirect object here—the person to whom the message is shown. This requires the Dative case, which is the focus of your current lesson! For masculine nouns and names ending in a hard consonant like Иван, we add -у to the end, making it Ивану.

Сообщение is the direct object (the thing being shown). Why didn't its ending change?

You are right that it is in the Accusative case. However, сообщение (message) is a neuter noun. In Russian, neuter nouns and their matching modifiers like это (this) do not change their form in the Accusative case. They look exactly like their Nominative (dictionary) form.

Why does the verb показала end in -ла?

This is the past tense form of the verb показать (to show). In the past tense, Russian verbs must match the gender of the subject. Since the subject doing the action is Она (she), the verb takes the feminine past tense ending -ла.

In English, we can say 'She didn't show Ivan this message.' Can we put Ивану right after the verb in Russian?

Yes, absolutely! Because the Dative case ending -у already tells us that Ivan is the one receiving the message, you can safely say Она не показала Ивану это сообщение. Russian word order is very flexible, and both orders are completely natural.