Modern chivalry is about returning to the "knight-like" qualities and incorporating them into your daily life. It doesn't mean paying for every dinner when dining in female company. In this day and age it is more chivalrous to shout your mate a meal, expecting nothing of it, other than the knowledge he'll probably appreciate a few dollars more in his wallet.
Bravery in your everyday existence is fundamental to chivalry. Bravery no longer means getting on a horse and going into battle without death crossing your mind, though. It means having the ability to confront pain, danger, or attempts of intimidation without any feeling of fear. You can be brave at work when you get into a situation with your boss, brave at home when you encounter family tragedy, and brave out on the street when you see somebody in trouble and decide to help them.
Courage, though often used synonymously with bravery, is more a state of mind that allows you to address and overcome the overwhelming difficulties in life, despite the unavoidable presence of fear. Courage becomes relevant, for example, when dealing with diseases such as cancer (your own, or those in those you love).
Courage can also mean standing up for yourself, or others, when put faced with emotional abuse. Emotional abuse can happen at home, or even at work with superiors: the fear of speaking up (because of the ladder-climbing/falling ramifications) is often what prevents courageousness. It is moral courage that motivates the best in society to call others out when they discriminate, whether directly or offhandedly. That includes pulling somebody up when they refer to something as "gay" when they mean "lame".
Although skill (as in military precision) was once essential in chivalry, today skill in any chosen trade or profession remains important in upholding this chivalrous trait. This means being the best at what you do, and continually up-skilling to ensure you adapt with modern developments and the competition. Just as a knight would be skilled in his swordsmanship through continual practice and training, the modern man should be skilled in his legal or medical knowledge, his roof-laying or metal-soldering abilities, or his creative writing or illustration skills.
Being generous in victory (and gracious in defeat) implies you will not let your successes get to your head, nor your failures get you down. Victory should be something you strive for and appreciate when it occurs (i.e. winning a big pitch at work or passing a big exam), but you should also expect that you will not always achieve victory. You must manage and learn from your defeats.
Of course, chivalry also still means being courteous to women. But that courtesy should be extended to all people around you. Whether that means helping your mother without being asked, or aiding your father with the tasks old age is preventing him from completing. It could mean giving up your bus seat to anyone that looks like they would appreciate a load off, or being there for your sisters when their partners are away (or for your brothers when they admit they don't know a hammer from a nail). In all, courtesy is being aware of one's environment and offering to help however possible.
In essence, modern chivalry is all about respect. Respect for those around you in every aspect of your life, and respect for what you do, and how you do it. So, in light of these modern explanations, ask yourself one question. How chivalrous are you?