Learn Polish: Mam na imię Piotr. A ty? - My name is Piotr. And you?

In Polish, you do not usually translate "My name is" word-for-word. Instead, you use the set phrase mam na imię, which literally translates to "I have on name." It is just the standard, everyday way to introduce your first name to someone!

Why is there no word for "I" before Mam?

The word mam already means "I have" because of its ending. In Polish, the verb itself tells you who is doing the action, so you usually drop subject pronouns like ja, which means "I", unless you really need to emphasize them.

Why is "And you?" translated as A ty? instead of I ty?

Polish has two common words for "and". You use i to simply add things together in a list. However, you use a to show a slight contrast, change the subject, or bounce a question back to someone else. Because you are shifting the focus from yourself to the other person, A ty? is the correct choice here.

Does imię mean my full name?

No, imię specifically means your first name. If you want to talk about your family name or last name, you need to use a completely different word, which is nazwisko.