После ссоры я решил извиниться перед подругой.

Because после requires the genitive case.

• Dictionary form: ссора = quarrel, argument
• Genitive singular: ссоры

So:

• после ссоры = after the quarrel / after an argument

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после фильма = after the film

It is genitive singular.

The base noun is ссора. Its forms include:

• nominative singular: ссора
• genitive singular: ссоры

So после ссоры means after the quarrel / after an argument, not after quarrels.

Context decides whether English is best translated as after the quarrel or after an argument.

Решил is the past tense, masculine singular form of решить (to decide).

So the sentence suggests the speaker is male:

• я решил = I decided (male speaker)
• я решила = I decided (female speaker)

Russian past tense often shows gender in the singular, which English does not.

The verb решить is perfective, and here it means to make a decision, seen as a completed act.

• решать = to be deciding / to decide in general, process-focused
• решить = to decide, to come to a decision

In this sentence, the speaker is talking about a single completed decision:

• После ссоры я решил... = After the argument, I decided...

That is why решил is the natural choice.

Извиниться is the perfective infinitive, meaning to apologize as a completed action.

Compare:

• извиняться = to apologize / to be apologizing (imperfective)
• извиниться = to apologize once, successfully complete the apology (perfective)

After решил (decided), Russian often uses an infinitive to show what someone decided to do. Since this is a one-time intended action, извиниться fits well:

• я решил извиниться = I decided to apologize

The ending -ся marks the verb as reflexive.

In modern Russian, извиниться / извиняться are the normal verbs meaning to apologize. You do not usually analyze it literally as to excuse oneself every time; it is simply the standard verb.

So learners should mainly remember:

• извиниться = to apologize (perfective)
• извиняться = to apologize (imperfective)

Russian uses a different structure from English here.

With извиниться / извиняться, Russian commonly uses:

• перед + instrumental

So:

• перед подругой = to my female friend / before my female friend

Literally, перед often means before / in front of, but with this verb it is the normal way to express the person to whom the apology is made.

Examples:

• извиниться перед мамой = apologize to mom
• извиниться перед учителем = apologize to the teacher
• извиниться перед друзьями = apologize to friends

Because перед requires the instrumental case.

• dictionary form: подруга = female friend
• instrumental singular: подругой

So:

• перед подругой = before/to a female friend

This is a regular instrumental ending for many feminine nouns ending in -а:

• мама → мамой
• сестра → сестрой
• подруга → подругой

It can mean either, depending on context.

Most basically:

• друг = male friend
• подруга = female friend

But in some contexts, подруга can also mean girlfriend. The sentence by itself does not force only one interpretation. If the meaning has already been provided to the learner as friend, that is perfectly normal.

Russian often omits possessives like my, your, his, etc. when they are obvious from context.

So:

• перед подругой can naturally mean to my friend
• Russian does not always need моей

If you want to make it explicit, you can say:

• перед моей подругой

But very often Russian prefers the shorter version when the relationship is clear.

The structure is:

• После ссоры = after the argument
• я решил = I decided
• извиниться перед подругой = to apologize to my female friend

So literally it is:

• After the argument, I decided to apologize to my friend.

This is a very typical Russian pattern:

time/context + subject + finite verb + infinitive phrase

Yes. Russian word order is flexible, although different orders can change emphasis.

The original:

• После ссоры я решил извиниться перед подругой.

A possible variation:

• Я решил извиниться перед подругой после ссоры.

Both are grammatical, but the original sounds very natural because it sets the time/background first: After the argument...

Russian often puts context-setting phrases at the beginning.

A careful stress guide is:

• После ссóры я реши́л извини́ться пе́ред подру́гой.

Main stresses:

• по́сле
• ссóры
• реши́л
• извини́ться
• пе́ред
• подру́гой

A rough English-style approximation would be:

• POS-lye SSO-ry ya re-SHEEL iz-vee-NEET-sa PYE-red pa-DROO-goy

But it is better to learn it from audio if possible, especially because Russian vowel reduction is important in natural speech.