Tobradex are medicated drops that contain both an antibiotic and also a steroid medication. As such, they are quite useful for treating both infection and inflammation.
When dealing with ears, Tobradex drops are most commonly used to treat one of two types of ear infections. The first is otitis externa, which is a condition in which the ear canal becomes infected and inflamed (commonly otitis externa is known as 'swimmer's ear'). The second is a middle ear infection
when the ear drum has become ruptured (when the ear drum is intact, the drops cannot penetrate to the middle ear). Given the fact that your doctor
said that there was something wrong with your ear drum, I suppose that, in your case, you have a ruptured ear drum with a component of infection.
You should use the drops exactly has they were prescribed. In particular, it is important to take them for as long as your doctor asked you to, even if the symptoms improved before then. You can expect that any pain, irritation, or discharge should begin to improve with these drops.
Make sure to followup with the doctor who prescribed the drops, especially if you don't notice any improvement.