Learn Italian: C'è un accordo tra me e te. - There is an agreement between me and you.

Because they come after the preposition tra (between). In Italian, after a preposition, you have to use special tonic pronouns like me and te instead of subject pronouns like io (I) and tu (you). This is similar to English, where we say "between you and me" instead of "between you and I".

Does it matter if I use fra instead of tra?

Not at all! tra and fra both mean "between" (or "among") and are completely interchangeable. You will hear Italians say fra me e te just as often.

Why is it C'è and not just È?

C'è is the combination of ci (there) and è (is). Just like in English, you need both words to say "there is" when stating that something exists. If you just used È un accordo, it would simply mean "It is an agreement."

Could I flip the order and say tra te e me?

Yes, grammatically both are perfectly fine. However, tra me e te is a very common phrase in Italian, so putting me first usually sounds a bit more natural to native speakers.