Learn Polish: Mój brat ma nowy parasol, ale zostawił swoją ładowarkę. - My brother has a new umbrella, but he left his charger.

The word swój (and its forms like swoją) is a reflexive possessive pronoun that means one's own. Because the brother is the subject of the sentence doing the leaving, using swoją makes it clear he left his own charger. If you used jego, it would usually imply he left some other man's charger.

Why is it swoją ładowarkę instead of swoja ładowarka?

It is in the accusative case because it is the direct object of the verb zostawił (he left). For feminine words in the accusative, the noun ending changes from -a to -ę, and the describing word changes from -a to -ą.

If nowy parasol is also a direct object being had, why do those words not change endings?

The word parasol is a masculine inanimate noun. For non-living masculine nouns, the accusative case looks exactly the same as the nominative (dictionary) form. Therefore, nowy parasol stays exactly the same.

Since the sentence is about my brother, why can't we say swój brat at the beginning?

The pronoun swój always refers back to the subject of the sentence, which means it cannot be the subject itself. Since the brother is the one doing the action, he is the subject, so we must use the regular possessive pronoun mój.