Learn Russian: Я потерял свой кошелёк!

Свой is a reflexive possessive pronoun that means "one's own". It always points back to the subject of the sentence. Since the subject is Я (I), свой means "my". While you could grammatically say Я потерял мой кошелёк, using свой is much more natural and preferred in Russian when the object belongs to the person doing the action.

Why is the pronoun exactly свой and not своё or свою?

The pronoun must match the gender, number, and case of the noun it describes. Кошелёк (wallet) is a masculine, inanimate noun. It is the direct object of the verb (the thing being lost), so it must be in the accusative case. For masculine inanimate nouns, the accusative case looks exactly like the dictionary form (nominative case), so свой doesn't change its ending here.

Does the verb потерял tell us anything about who is speaking?

Yes! In the past tense, Russian verbs change their endings to match the gender of the subject. Because потерял ends in -л, we know the speaker is male. If a woman were saying this, she would add an -а to the verb and say Я потеряла свой кошелёк!

Do I have to include Я at the beginning of the sentence?