Learn Italian: Se vincessi alla lotteria, sarei un uomo molto fortunato. - If I won the lottery, I would be a very lucky man.

This is the second conditional, used for hypothetical or unlikely situations. In Italian, the 'if' part of a hypothetical sentence (se) must use the imperfect subjunctive tense, which is vincessi here. The result part uses the present conditional (sarei).

Vincessi is both the 'io' (I) and 'tu' (you) form in the imperfect subjunctive. How do we know it means 'I won' here?

You have to look at the other half of the sentence for context. The conditional verb sarei strictly means 'I would be'. If the sentence was about 'you' winning, it would use the 'tu' conditional form: se vincessi alla lotteria, saresti... (if you won, you would be...).

Why does it say alla lotteria? Doesn't vincere mean 'to win', so shouldn't it just be vincere la lotteria?

In English, we win a thing directly ('win the lottery'). But in Italian, when you win a game, sport, or contest, you usually use the preposition a. So 'to win the lottery' translates to vincere alla lotteria (literally 'to win at the lottery').

Could I swap the two halves of the sentence around?

Yes, you can completely reverse the order without changing the meaning or the verb tenses: Sarei un uomo molto fortunato se vincessi alla lotteria.