Ich schreibe eine kurze Nachricht an meine Schwester.

Because ich takes the 1st person singular form of the verb in the present tense.

The infinitive is schreiben = to write.
In the present tense:

• ich schreibe
• du schreibst
• er/sie/es schreibt
• wir schreiben
• ihr schreibt
• sie/Sie schreiben

So Ich schreibe means I write or I am writing.

Because Nachricht is a feminine noun: die Nachricht.

Here it is the direct object of the sentence, so it is in the accusative case. For a feminine noun in the accusative singular, the article stays eine.

The adjective kurz also has to match the noun pattern, so it becomes kurze:

• eine Nachricht
• eine kurze Nachricht

So:

• eine = feminine accusative singular article
• kurze = adjective ending that fits that pattern
• Nachricht = feminine noun

Because all German nouns are capitalized.

So in this sentence, both of these are capitalized:

• Nachricht
• Schwester

That is one of the most noticeable spelling rules in German.

With schreiben, German often uses an to show who something is being written to or addressed to.

So:

• eine Nachricht an meine Schwester schreiben
• einen Brief an meinen Freund schreiben

In English, you say to my sister. In German, the natural wording here is often an meine Schwester.

Because an takes the accusative here.

Schwester is feminine, and with mein- the feminine accusative singular form is meine.

Compare:

• nominative: meine Schwester
• accusative: meine Schwester
• dative: meiner Schwester

So in this sentence, meine Schwester is correct because the phrase is an + accusative.

Yes, absolutely.

That version is also very natural. In that sentence, meiner Schwester is in the dative case and works like an indirect object.

So both are correct:

• Ich schreibe eine kurze Nachricht an meine Schwester.
• Ich schreibe meiner Schwester eine kurze Nachricht.

The meaning is basically the same. The difference is mostly in structure:

• an meine Schwester = uses a prepositional phrase
• meiner Schwester = uses a dative indirect object

German allows both patterns with schreiben.

Because in German, attributive adjectives usually come before the noun, just like in English.

So:

• eine kurze Nachricht
• ein langer Brief
• mein neuer Computer

The adjective also changes its ending depending on the article, gender, number, and case. That is why it is kurze, not just kurz.

Yes, but the conjugated verb still has to stay in second position in a normal main clause.

The basic sentence is:

• Ich schreibe eine kurze Nachricht an meine Schwester.

But you could also say:

• Eine kurze Nachricht schreibe ich an meine Schwester.
• An meine Schwester schreibe ich eine kurze Nachricht.

In each case, schreibe is still the second element in the sentence. That is a very important German word order rule.

A few useful points:

• ich: the ch is the soft German sound used after front vowels
• sch in schreibe and Schwester sounds like English sh
• ei in schreibe sounds like English eye
• z in kurze sounds like ts
• w in Schwester sounds like English v
• ch in Nachricht is not like English ch in church

So this sentence includes several very common German sound patterns.