Sometimes it is very difficult to do what you feel is right in spite of what the rest of the world is doing. The rest of the world, or your group, may mock you, and criticize your beliefs, so it can take real bravery to stand up to the crowd. This is what the quote means. Literature is full of characters who were forced to stand up for their own beliefs and obey their consciences instead of following the crowd. John Proctor in "The Crucible", Atticus in "To Kill a Mockingbird", Hester, in "The Scarlet Letter", Antigone in "Antigone", John in "Brave New World" and Montag in "Fahrenheit 451" are just a few literary characters who are brave individuals who follow their conscience and deal with the consequences of that decision.
I think that the beauty of this quote is that you can interpret it however you want. It can be as meaningful as you want it to be, and I think that for most it is pretty important to them, including myself.
To me, this quote means that you (I) can do whatever you want to do with your life. You shouldn't let people tell you what to do, and don't let them influence you either. You should pave your own path, go against the grain, and show the world that you are an awesome individual. Don't simply do as you're told, question why and if you don't like it, find an alternative.
Hope this helps!
It means that it is a very courageous thing if you break apart from the societal norms and follow your own heart. I'm sure Atticus in To Kill a Mockingbird is one of many examples because Atticus is taking the side of justice which is also the side of an African man. Because of this he received a lot of hate, but he followed his heart and did what was right.
It means that it takes a great deal of courage to do what your conscience tells you to do rather than following the crowd. Your conscience tells you what is right and what is wrong, and sometimes following through with what is right comes with the possibility of having to stand up to others, even your friends, by refusing to do the wrong thing, often the more popular thing. That takes a great deal of bravery.
Refer to Robert Frost's "The Road Not Taken" for a more poetic answer.
Good luck with your assignment!
One's own conscience can often go against the flow of public opinion, action, or what is considered acceptable. If you follow your own conscience rather than that of the majority, you will be alone many times. This means you will have to defend yourself and your actions, possibly at the outrage or contempt of others. However, not swaying from what you truly believe to be right, despite the attacks or pressure from others is what defines you as truly brave.