Chivalry is an incredibly complex creation, to properly answer a question such as this would require a mammoth reply, but instead I would direct you to certain historians for a complete answer. Maurice Keen has done brilliant work, his book simply titled 'Chivalry' will offer a great deal of insight you may find very intriguing. In a similar vein of broad works on chivalry; 'The Knight and Chivalry' by Richard Barber is also very good, and was very useful during my studies. My particular work led me to look indirectly at how chivalry played a role in so many of the actions of the knightly classes, and so I will comment below on some of this, particularly in the later medieval period.
It is important to consider that chivalry was intrinsically tied to the knightly class, and would transcend borders or rivalries. Throughout large periods of the Medieval period it would be instrumental in driving the actions of the nobility across Europe. Individuals could, and did, make careers out of being known as a chivalrous knight, but equally actions that could be attributed as due to chivalry could in reality be down to other factors, such as a desire to travel, to see the world and to simply have fun. The Reisen of the Teutonic Order are a fascinating example of this. Undoubtedly these expeditions from the Baltic to fight against the pagans of Lithuania gave real kudos to participants, particularly from a chivalric perception, and as such chivalry was a driving factor, but as is ever the case so much more was at play. People joined expeditions like the Reisen for many reasons; chivalry was certainly a driving force, but so was a desire to advance their career or to follow in a family member's footsteps. Frequently we see branches of the nobility visiting the Teutonic Order and joining the Reisen because of family connections, perhaps due to a member of the family joining the Order and inviting them over for instance, and actions like this could distinguish an entire family as notably chivalrous.
An excellent example of the influence of chivalry is the French Marshal Boucicaut. Boucicaut came from a privileged background, and his father before him had been Marshal of France, highlighting again the continuation of family. Boucicaut was renowned throughout Europe; if there was a chivalrous activity somewhere it was quite likely he had done it. Boucicaut crusaded in Granada, he joined the Reisen in the Baltic, he fought against the Turks in Bulgaria, and he visited many of the monarchs of Europe. He even went so far as to travel to Egypt and offer his services to the Mamluk sultan. Much of this can be found in his biography the 'livres des faits' (Book of Deeds), but it is entirely presented to make Boucicaut look good, written presumably by a member of his household, yet it still offers a remarkable view of an individual driven and renowned by chivalry. The sheer amount of Boucicaut's travels marks him as remarkable, but individually they were fairly common for the male aristocracy. For Boucicaut, these actions made him just about the most famous knight of his age, even with his connection to disastrous defeats such as the battle of Nicopolis and Agincourt. It has often been said as well that the Knight in Geoffrey Chaucer's 'the Canterbury Tales' was modeled as an English version of Boucicaut, although interestingly Terry Jones of Monty Python fame argues Chaucer's Knight is instead used as a satirical figure to mock chivalry.
I realise I'm beginning to write a rather long entry so I'll end it here, hope it is of use.