What is the best way to correct an overbite without surgery and without braces that are visible in front of the teeth?

There are many options to correct overbite depending on its severity, but I should agree with my colleagues here: a proper examination, evaluation and diagnostic work up is needed to determine the best option or the option that would fit your particular situation. See an experienced cosmetic dentist, and get as many second opinions including one or two from your local orthodontist to learn more how you may need to proceed. Good luck !!

Without knowing the details of your situation, I will give a general answer.  Overbites can sometimes be corrected with crowns or veneers.  This can be very invasive and costly.  There are orthodontic techniques that hide the brackets and wires behind the teeth.  Clear aligners (Invisalign, Clear Correct) are almost undetectable.  You can also use clear or tooth colored brackets and wires.  Often, there is some combination of prosthetics and orthodontics to get an ideal result.  You will need to consult a dentist. 

The questions you are asking can only be answered by a dentist at a consultation.  They should be able to give you some answers after seeing your situation.