Translation:I want to invite you to my birthday.
That's just the way we say it in Ukraine. We just запрошуємо на день народження and then give all the details. It maybe a party or just birthday ice skating :)
We don't necessarily call that вечірка at all. Wait, we don't even use the word "вечірка" that much! :) If it's gonna be a party (with quite a few people) at the club, than it's likely to be called вечірка. But it's not for small house parties for under ten people (and that's what most birthday celebrations here look like)
We usually just say "піти у гості". That literally means something like "to visit someone" (гість is "visitor"), but that usually does mean that that someone had some kind of house party or dinner. You can say "Учора я ходив у гості до мого брата" or "Учора я був у гостях у мого брата" (Yesterday I "visited" my brother). When you invite someone you say "Запрошую вас до мене у гості" or "Приходьте до мене у гості"
In (UK) English we will often say just birthday with the party/event part implied, I assume it's just the same here. It's a lot more common to say "are you coming to my birthday?" than "are you coming to my birthday party?" It also means it can be applied to all sorts of birthday related gatherings that party might not sound quite right for.