صديقتي كتبت الرسالة في المكتبة قبل الدرس.

It comes from صديقة plus the possessive suffix ـي, which means my.

So:

صديقة = a female friend
صديقتي = my female friend

When a feminine noun ending in ة takes a suffix, that ة is pronounced as t, which is why you get صديقتي, not صديقةي.

In Modern Standard Arabic, صديقتي normally just means my female friend.

It can sometimes be understood romantically from context, but by itself it is not necessarily girlfriend. Native speakers usually rely on context, or use other wording if they want to make the romantic meaning completely clear.

Because the subject is feminine singular.

كتب = he wrote
كتبت = she wrote

Since صديقتي refers to a female friend, the verb must be feminine singular in the past tense: كتبت.

Arabic verbs already contain subject information.

So كتبت already means she wrote. You do not need to add هي unless you want extra emphasis, contrast, or clarity.

Yes. Both orders are possible in MSA.

This sentence uses subject + verb order:

صديقتي كتبت الرسالة...

But you could also say:

كتبت صديقتي الرسالة...

Both are grammatical. Starting with صديقتي makes my friend the topic or point of focus.

الرسالة is the direct object of كتبت. It is the thing that was written.

Also, the prefix الـ is the definite article the, so:

رسالة = a letter / a message
الرسالة = the letter / the message

In full grammatical terms:

صديقتي is nominative
الرسالةَ is accusative
المكتبةِ is genitive
الدرسِ is genitive

In normal Arabic writing, short vowel endings are usually not written, so learners often have to infer the case from the grammar of the sentence.

في is the preposition in.

After a preposition, the following noun is normally genitive. So in a fully vowelled version, this would be:

في المكتبةِ

That whole phrase means in the library.

قبل means before here.

It is often treated as an adverbial noun in Arabic grammar, and when it is followed by another noun, that noun comes in the genitive:

قبل الدرسِ = before the lesson

So الدرس is genitive because it is linked to قبل in this construction.

Because ر and د are sun letters.

With sun letters, the ل of الـ is not pronounced. Instead, the next consonant is doubled in pronunciation.

So:

الرسالة is pronounced ar-risāla
الدرس is pronounced ad-dars

This is a pronunciation rule, not a spelling change.

You may see it explained both ways.

Because it starts with صديقتي, many grammar books analyze it as a nominal sentence whose predicate is the verbal clause كتبت الرسالة في المكتبة قبل الدرس.

For a learner, the most useful practical idea is simply this: it means My friend wrote the letter in the library before the lesson, and Arabic allows this subject-first order.

A careful fully vowelled version is:

صَدِيقَتِي كَتَبَتِ الرِّسَالَةَ فِي الْمَكْتَبَةِ قَبْلَ الدَّرْسِ

One small detail: in connected reading, كَتَبَتِ has a kasra on the final ت to avoid a difficult consonant cluster before الـ. If you stop after the verb, you would pronounce it as كَتَبَتْ instead.