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I'm not sure where your confusion is.
In American English, the thing you're handed at the end of a meal is often called a check. It is a synonym with bill. That piece of paper is called a cuenta. In every Mexican, Colombian, Ecuadorian, and Cuban restaurant I've been in, they call it "la cuenta" and it's what we call "the check".
This is not referring to a check that I write out (which draws money from my checking account). Some people in the US spell it cheque and it seems cheque is the only spelling used in the UK.
está (with the accent) is a form of the verb, Estar (to be). (yo) estoy - I am; (tú) estás - You are; (él/ella/usted) está - He/she/you (formal)/it is; (nosotros) estamos - We are; (ellos/ellas/ustedes) están - They/you (all) are
So in this example, you're saying "Here (it is) the check". So, we use está.
Esta, without the accent, means "this", and is used when referring to a feminine object. Este is its counterpart for "this", reffering to a masculine object. "Esta biblioteca tiene muchos libros", for example. This library has a lot of books.