"We are inviting you to our house."
Translation:Zapraszamy cię do naszego domu.
I'm sorry, but I have to disagree - the answer is closer to what MattMattM suspected. "do" would work with activities ("zapraszam was do wspólnego śpiewania" = I am inviting you to sing together), and then it also works just like in "You are going to our house", which is a closed space.
However, if I invited you to an airport, or a post office (let's remember that the 'closed space' rule is not a very solid rule), that would be "Zapraszam cię na lotnisko/pocztę", just like with going.
And for something that is actually very common with inviting - events. They take "na + Acc" when you are going to an event, same with inviting. Zapraszam cię na koncert/urodziny/imprezę.
They are both genitive=accusative of Ty, but Cię is not accented, and Ciebie is accented At the beginning of the sentence and after preposition it has to be Ciebie, in longer sentence if preposition is at the end it is accented, so also ciebie (and by longer I mean sentences when you can put ciebie in the middle). If cię/ciebie is in the middle of sentence you can use either, but cię is not accented and ciebie is accented.
this happens with all prepositions that have more versions in the same case. (he/she/it/they have third n+ version that are used after prepositions)