Heute Morgen habe ich der Nachbarin einen Brief geschrieben.

The verb schreiben requires two objects:

• an indirect object (the person you write to) in the dative
• a direct object (the thing you write) in the accusative

Since Nachbarin is feminine, its nominative die Nachbarin changes to der Nachbarin in the dative singular.

Look at the roles:

• der Nachbarin is the recipient → indirect object → dative
• einen Brief is the thing being written → direct object → accusative

You can also spot it by the articles:

• der (feminine dative)
• einen (masculine accusative)

General word‑order tendency when both objects are noun phrases:

1. Dative object (indirect)
2. Accusative object (direct)

So … habe ich der Nachbarin einen Brief geschrieben is the neutral order.
You can swap them for emphasis on the letter:
“Habe ich einen Brief der Nachbarin geschrieben”
—but it sounds unusual in everyday speech.

Heute Morgen is a time adverbial placed in the “prefield” (first position) for emphasis. In German main clauses, the finite verb must be in second position, so when you put Heute Morgen first, habe follows immediately:

1. Heute Morgen
2. habe
3. ich …

You could also say:

• “Ich habe heute Morgen der Nachbarin einen Brief geschrieben.”
• “Ich habe der Nachbarin heute Morgen einen Brief geschrieben.”

German typically orders adverbials as Time – Manner – Place.

This is the Perfekt (present perfect) tense structure in German:

• The finite auxiliary (haben or sein) occupies the second position in a main clause.
• The past participle (geschrieben) goes to the end of the clause.

In spoken and informal German, Perfekt is the default way to talk about past events.

Yes. In writing or formal contexts you might prefer the Präteritum:
“Heute Morgen schrieb ich der Nachbarin einen Brief.”

Differences:

• Perfekt (habe … geschrieben) → common in speech
• Präteritum (schrieb) → common in narratives, reports, formal writing

Yes. You can rephrase the indirect object as a prepositional phrase with an + accusative:
“Heute Morgen habe ich einen Brief an die Nachbarin geschrieben.”
Here:

• einen Brief = direct object (accusative)
• an die Nachbarin = prepositional phrase indicating the recipient

Use the dative pronoun ihr (“to her”). Pronouns generally come before noun phrases:
“Heute Morgen habe ich ihr einen Brief geschrieben.”

Rule for two pronouns:

• If both objects were pronouns, the accusative pronoun comes before the dative pronoun (e.g. “Ich habe ihn ihr gegeben.”).

The usual sequence is:

1. Time (Wann?)
2. Manner (Wie?)
3. Place (Wo?)

In our sentence we only have a time expression (Heute Morgen), so it naturally appears before any manner or place phrases. If you added a manner adverbial, it would come after the time and before any place adverbial.