When you "submit" an application, a homework assignment, or some other official document, it means that you give it to the person or organization who has asked for it and who has authority over it. For example, you "submit" a job application to the company that might hire you.
The word "submit" is a slightly formal word. In casual conversation, you can use the phrase "turn in (something) or "turn it in":
Make sure to turn in your application by Friday.