Fofa's and Bergman's answer are perfect. Here's a few other ways to say the same thing: "to do my best" means: - to do the best that I can do - to try as hard as possible - to make every effort to do something well - to use all my skills, knowledge, ability to do something well. And that's the best that I can do. I've tried my best to answer your question.

I was said once that in fact it was not as strong as it may appear. It assumes "withing the limits of our contract/ my duty etc". The person saying it, doesn't take it as a personal obligation. That's why if you want to say that you are really committed to do something, you need to use another expression. Otherwise, it may look insincere and not reliable.

I'll do my best to give you an example ;-)

do my beat means to try as hard as possible i'll do my best in the test to get the full mark. sam said that he will do his best to finish his work early and be here on time.

It means to strive for the highest.