It seems in watching old movies or reading about history that the sense of honor has been very important in terms of it's ability to shape the actions of individuals or cultures (people were worried about how their actions influenced their honor). I feel as though this sense is generally lost in today's society (at least in the west, I cant speak for eastern cultures). Nowadays, people dont worry about how their actions affect their honor, nor does the general population consider how honorable an individual is, at least not in such direct terms. Has the term "honor" simply become obsolete, or replaced by some similar principle? Is it too vague a notion for todays society? How has the sense of honor evolved over the last 100 years in American/western culture?
I'm certainly no expert, but I'll give my point of view.
The definition of "honor" in the western Euro-American tradition, at least 100 years ago, was in many ways connected to the romanticized notion of medieval chivalry. This concept was assumed to be a guideline for gentlemanly and morally just behavior that combined elements of honor, loyalty, piety, and etiquette among other virtues. There is some dispute to how much emphasis was given to the notion of chivalry in the middle ages among scholars today, but it can be said by the early modern era; (1492-1815 a.d. roughly) and well into the 19th and early 20th century that a pseudo-chivalric, or proper "gentlemanly" behavior was well established, especially among the elite and noble class. Just reading Dumas's; "The Three Musketeers" one has an almost immediate sense of the importance of honor, loyalty, and reputation; in which the characters constantly duel each other over what would today be considered slight offenses such as verbally disrespecting one's family name, or accidentally bumping the shoulder of a senior officer, etc. Although this is fictional, and may be a satire, I think that it is based of of some truth; in which one's honor and reputation was considered very important, important enough to die for. Dueling became quite common in the 18th and 19th centuries among men of the aristocracy and leading classes, so much so that many countries went through the trouble of officially outlawing it and establishing it as its own separate crime.
As your question to its decline in personal importance, I think it has several reasons. The first I believe is its preconceived association with the nobility and elite. During the 19th century, expanding on ideas of the Enlightenment and Revolutions of the previous century, an interest of the lower and working classes grew in forms of new movements such as Socialism/Marxism and Anarchism etc, and old ideas of noble, chivalric, honor, became to be interpreted as archaic, arrogant, and even despotic in a sense, associated with the old regimes; with some groups rejecting any form of loyalty and servility. However, there were counters to this in the form of National Romanticism,Conservative Royalism, and later Fascism which encouraged a strong sense of honor and loyalty. With honor becoming associated with the elite and noble classes of the old regimes, it inevitably became embroiled with politics. However, chivalric honor in a sense would continue throughout the end of the 19th century, partly due to the rise and domination of Nationalism. The second cause of its decline however, I would associate with the World Wars, especially WWI, when much romantic, or chivalric ideas about war were completely disregarded in the face of the incredible slaughter. This trend continued in a sense with the rise and fall of the Nazi's and Imperial Japan, which were both extreme right-wing entities that held traditional ideas of honor, loyalty, and duty in high esteem. The third reason for its decline is more in regards to honors personal connotations. Which, in the US at least, the rise of youth culture has an impact. The counter-culture of the 1960's and dissatisfaction of the Vietnam War led to further questions of traditional ethics and etiquette. Also popular music, such as rock n' roll, which initially evoked a sort of rebellious, anti-authoritarian lifestyle became heavily popular to the youth of the 50's and 60's in which a libertine, rebel lifestyle began to be seen as "hip" or "cool". This trend can be seen with popular culture as a whole as well, such as fashion where suits and dresses gradually were replaced by jeans, T-shirts and other casual wear in nearly all occasions except formal or business events.
In all, I think the "honor' you mention describes this "chivalric" notion explained above. Except in the most formal of occasions, yes I think this concept of honor is dead in the west. Unfortunately, that behavior has been linked to politics, and for many is seen as something that is "conservative" or "traditional" or even out-dated. Generally, I think it would be strange for an average person today to regard their personal honor or reputation as a top priority, much less I think anyone would be willing to die for it. Even in today's military's the emphasis has changed from personal honor and glory, to protection and salvation.