It means that you can identify with the character, their experiences, what they are going through. It has more to do with what you are or what you have experienced, than what you would like to be. Sometimes you relate to the character exactly because they are making the same mistakes as you tend to do, or exhibiting your the qualities you have that you would rather change.
As in, Data's burden of potential immortality naturally leads to a desire to grow old and die, which speaks to my personal experience? Or, looking at it in another way, Harry Kim's ordinariness and lack of cool is so unlike my personal experience that it just can't resonate with me?
People
say things like the quote above but they often have no discernible connection to the opinions offered. The original post talked about realism but somehow managed to conclude the most unrealistic (therefore, unrelatable) character in Trek was Kathryn Janeway? Spock, Data, Odo and Seven of Nine are not even supposed to be ordinary even in their own (science)fictional universe!
"Relatable" has absolutely nothing to do with
ordinary. It has everything to do with the character having recognizable human emotions, reasoning and reactions, which make the viewers empathize with, or at least recognize where they're coming from. And that is very difficult when the character is written inconsistently and as a Mary Sue-captain who seems to be motivated mostly by what the writers want to happen in a specific episode. or which decision they think the captain should make in the episode so that she would be right once again. It is also very difficult to relate to a character who is written without any real depth, and whom we never really get to know well.
(edit: But I do relate with Janeway on one thing - I love coffee
)
There is such thing as psychological realism. And yes, it means that, in SF, androids, aliens and holograms may actually end up being far more relatable than many human characters - especially when the writers use those non-human characters to explore issues such as loneliness, social ineptitude, being confused by one's emotions, feeling like an outsider, needing to belong somewhere. These are the things that many people can relate to. Heck, most teenagers (or those who remember what it is like to be one) will very easily identify with all this.
In the example of Kira Nerys, I have no idea how anyone could find her more relatable on the grounds of superior realism, as the OP would have it. She was supposed to be simultaneously a slick undercover operative/terrorist and a bluff, straightforward soldier (despite not being in the army!)
Right, because members of resistance movements in occupied countries were all seasoned officers who had finished military academies.
edit: Also, what undercover operative? The closest we've ever seen to this is when she pretended to be a prostitute for like 2 minutes so she could get close enough to the guard to punch him out (in "Homecoming"). That must have really required extensive undercover skills.
When was it ever mentioned that she had an undercover identity and task? Unless you're mixing her up with Iliana Ghemor.
She was supposed to be devout but didn't have theological interests
Because every person who believes in God or has any other kind of religious belief is an educated theologist who studies the works of St Augustine and Thomas Aquinas.
She was supposed to be studiously disinterested in politics yet somehow a revolutionary/terrorist. She might have realistically been some of these things, but she can't realistically be the opposites simultaneously.
Because all the members of resistance movements in occupied countries, say in World War II, or in any country under the Ottoman Empire, or in Mexico, or any country under colonial rule, are always staunch communists or capitalists or believers in some other ideology, and had decided to fight in order to help a certain political party come to power - rather than because they felt they had to protect themselves and their families, or because they found the conditions under which they lived unbearable, due to hunger, oppression, cruelty or abuse.
Some of the people in my extended family on mother's side had fought in resistance/guerilla movements during the World War II, and guess what one of them told my mother once? That they didn't know or care about communism or royalism or anything, they just went to the woods and took up arms because they had to. Because the Nazi-satellite state that included the territory where their villages were, had a genocidal policy towards their ethnic group. It was pretty simple - either you were going to get literally slaughtered or have your eyes plucked out before getting killed or taken to a concentration camp, or you were going to run away to another country as a refugee, or you were going to fight. Ideology had nothing to do with this, it was a matter of survival. It was only later that the political commissaries came and started organizing them into a broader movement and talking about anti-fascism and communism, but most of those people, who ended up as members of communist resistance movement against the fascist occupation, were just ordinary people, and ideology was a side issue for them.
Her love life went from one big shot to another, culminating in the key figure in galactic war.
Because it's totally unheard of for a public figure to be involved with a series of celebrities/public figures. I can't think of this ever happening in real life.
She was basically a nice Jewish girl whose mommy was Hitler's lover. (Yes, not literally true!)
Certainly not, since the parallel doesn't work. As it has been correctly noted in another thread in General Trek very recently, the Cardies/Bajorans - Nazis Germans/Jews comparison is made of fail. If Cardassia had had a widespread genocidal policy towards the Bajorans, you can be sure that there would have been far more than 10 millions Bajoran victims over the course of 50 years. The Hutus were more efficient in killing Tutsies. And on DS9, the Dominion managed to kill at least 600 million Cardassians in just a few weeks. Cardassians considered Bajorans an inferior, superstitious, culturally backward race, but they did not see Bajorans as the bane of their existence - at least not before the Resistance actions escalated - and did not decide on a Final Solution to their problem; they did not occupy Bajor because they wanted to kill all Bajorans - they did it primarily to plunder their natural resources, and secondary, to use the people as slaves and exploit them as work force or sexually. Sounds familiar? A few Cardassians, like Gul Dar'heel, did in fact commit genocide, but the main Cardassian policy, as exemplified by Dukat, the Prefect of Bajor, was very much White Man's Burden (Grey Man's Burden?). In other words, the more adequate comparison are not Nazis, it is colonialism.
And I wonder, just how many girls in the colonized worlds had mommies who slept with their masters? Again, unheard of...
Obviously, Kira has her fans, but it's not because she's realistic or relatable in the sense above.
In what sense? She is very realistic and relatable, because she behaves and feels like you'd expect from a person with the history she's supposed to have.
Saying that an SF show should be realistic in the sense of showing everyday life of an average person would be ridiculous. But SF shows can be as psychologically realistic as any other show, and they don't have any more excuse for a lack of psychological realism than any other show.
Travis Mayweather was memorable, at least, because the fact that he never had anything to do was discussed all the time. But Reed? Seriously: What was the point? 
