The answer here is interesting. The Ancient Romans did not vote like individuals as we do today; instead the voted in "Tribal Assemblies." Obviously, each Roman Citizen belonged to a family, but it is not a family as we would think of it today except in the broadest sense. "Gens" was the word they used and it is often translated more as "Clan" than family. Each "Gens" would have a nominal leader and it was commonly a person of fairly high status in the community, even in the case of Plebian Gens. That person (always a patriarch) would be responsible for keeping lists of the members in his Gens. The Romans were extremely conscious of their lineage and often directed prayers and sacrifices to illustrious ancestors. See Oxford Classical Dictionary. http://classics.oxfordre.com/abstract/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-2820?rskey=exbXUw&result=1
So that is the "Gens" - but above the Gens would be the "Tribus" or Tribe as a super-group containing multiple extended families. In Roman legend, the first 3 Tribes were created by Romulus, along with the founding of Rome itself. Eventually, by the late Republic and Early Empire, there were some 35 tribes. http://classics.oxfordre.com/abstract/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-6555?rskey=SNobYj&result=1
I could go on about this in some detail, but the fundamental part which answers your question is that the Tribes were responsible for assembling their voters for "Comitia" (whence our word "committee") which was an assembly, or group of assemblies, of adult male Roman citizens for a specific purpose, such as electing a Tribune - the "Comitia Tributa." When the omens or "auspices" were favorable, the election would be held - be it for Tribune of the People, etc. The Comitia would be overseen by a magistrate, and the majority vote of the Comitia would then become the vote of that entire Comitia. In this way, the large number of Roman citizens could be divided into a number of Comitia, organized according to their Tribe. The vote of each Comitia would be recorded, and the majority vote would prevail.
Remember of course that the commoners or "plebs" could only vote for Tribunes for most of Roman history. Only Patricians were eligible for membership in the Senate.
Again,See Oxford Classical Dictionary. http://classics.oxfordre.com/abstract/10.1093/acrefore/9780199381135.001.0001/acrefore-9780199381135-e-1747?rskey=yk3vxD&result=1
Although I am frequently skeptical about the quality of Wikipedia, as everyone should be, there is actually a pretty good article on this. They have a good diagram which may help you to understand the process even better. It also details the different types of Comitia and the purposes served by each. https://en.wikipedia.org/wiki/Centuriate_Assembly