There's alot of roles, the firstmost, and primary one, is voting. Only a citizen can vote.
This was more prevailent in previous eras where women and slaves, blacks, etc, weren't allowed to be citizens, as they were not considered 'real people', but rather 'subhuman'. As such, during this timeframe, citizenship denoted who had the right to vote.
Beyond that, citizens have several other 'duties', as they are assumed by default to be more loyal and patriotic. They are assummed to be willing to fight for their country in one way or another, either by the armed forces, or via militia, geurilla warfare, or other methods, and they are also assumed to defend it from internal threats as well.
A citizen also pays their taxes to support their countries' infrastructure, and because of this, and their other duties, these are the reasons they're allowed to vote, as it's assumed only they have the willingness to contribute enough to their country, so only they deserve to participate in the choice of such.
This's since been changed over the years, and now doesn't really mean much of anything. A 'citizen' of a country mostly just means yeu're allowed to vote and have a passport. It more or less states yeu're legally allowed to be there as well. Illegal immigrants are not considered citizens.
Beyond that, it may have many other terms but it's hard to find much information so who knows.
In the 'utopian' (not really) future of starship troopers, they make referance to citizenship several times, though mostly in a military sense. I wouldn't pay much attention to such as it's actually an outdated method of belief, not a futuristic one at all.