We hear some version of kaddish more often in public worship than any other prayer. There are at least five forms of it and countless variations on those five from community to community. In its original form, the kaddish was used to mark the end of a study session, and it was recited by the teacher. It evolved into what we now call kaddish d'rabbanan. It was also recited in the house of mourning for a scholar, generally by his students who would gather to study in his memory.
Families of the less-educated complained about what they perceived as the lack of honor due their late beloved. As such, the custom of reciting an abbreviated form of kaddish d'rabbanan emerged. It became known as kaddish yatom, the Mourner's Kaddish, recited by family members at the end of daily services. Perhaps because of a legend that Rabbi Akiba rescued a soul from gehenna by reciting kaddish, mourners began reciting it on a daily basis after a death.
Because the kaddish is in Aramaic, the street language of the early centuries of the common era, it was well-known. Because it has a rhythmic quality and magnificent sentiments about God's grandeur and our hopes for God's presence, it was also very popular. And because it became so closely associated with end points, it was adopted into daily worship in two forms -- kaddish shalem (the Full Kaddish) and chatzi kaddish (the Half Kaddish) which function as punctuation in the flow of the service, as if they were period and semi-colon, respectively. Similarly, an extended kaddish recited exclusively at burials developed.
In all cases, it is recited only in the company of a minyan.
The 16th-century code Shulchan Arukh notes two things about standing for kaddish. The first is that whoever is reciting it must stand. The second is that there are communities in which all congregants stand for every kaddish, and others in which they do not (unless the kaddish follows a standing prayer, like Hallel). The custom is local.
In most Conservative synagogues, the former custom is the most usual. Only those reciting kaddish rise for its recitation. In most Reform and some Conservative congregations, everyone stands for Mourner’s Kaddish in a gesture of solidarity with those who are bereaved (but not for the other recitations, as a rule). Our custom has always been for mourners only to rise.
Some congregants have suggested that I invite everyone to rise for Mourner’s Kaddish as a means of consolation for the immediately bereaved and a show of memory for the martyrs of our people. Some have strong aversions to rising at that moment because of familiar custom, including the folk wisdom that one does not stand for kaddish while a parent is alive. Some feel that the public proclamation rising above the seated crowd is important to their own process of grieving and healing.
One may rise for any and all recitations of kaddish, but a requirement to do so would likely make many people uncomfortable. I will work on some appropriate language to make the option clear; in the meantime, you are appropriate in whatever custom you observe.