From a physician's perspective, a person is mentally ill if they're failing at life, and their mentations are to blame.
Every society has standards that it teaches its young and expects is citizens to follow. This includes a lot of things that people are expected to, from an appropriate age on, do for themselves with minimal help. When a person fails to reach a milestone, or reaches it and then regresses back, society labels this a problem, and deems the person in need of help to correct that problem and [re]reach that milestone. For example, in the USA, it is considered problematic and unacceptable to start neglecting personal hygiene, or to stop caring about finding gainful employment. When this happens, doctors, psychologists, social workers, and other professionals called in on the case examine the person and his life in search of a culprit. When the person's pattern of mental activities appear to be driving the unacceptable behavior, he is deemed to have a mental illness, and efforts are focused on eliminating or correcting the mentations that appear to be leading to the unwanted behavior patterns. These efforts can involve drugs, but just as often not, depending on what the mentations (and resultant behaviors) are, and how deeply ingrained they are in that person.
The point is that the notion of mental illness is based upon a concern for the greater societal good, and different umbrellas of mental illness reflect different philosophical ideas about what constitutes a good life, what constitutes an ideal society, and how an individual ought to fit into his society.
The same can really be said for physical illness. Someone is physically ill if their bodily state poses significant hindrances to their performing tasks that their society expects all people to be capable of. The controversy with mental illness comes from the fact that other people's physical states are readily observable and measurable, while their mental states are not -- all we can observe is behavior, which we assume correlates reliably to mental states.
Obviously there's much room for debate, because every society is a complicated, heterogeneous, dynamic entity. What one person considers normal and standard may not be what others feel. So we clinicians tend to boil it down to three questions:
1. Do your mental activities distress you?
2. Does the way you conduct yourself tend to distress others you encounter in your day to day life?
3. Does this distress diminish your quality of life, relative to others around you?
If yes to two of these three, then you need help. Otherwise, you are not mentally ill; you are simply experiencing life in all its painful glory, with your particular cross[es] to bear. This is not to say you're not in need of a listening ear, or that you don't deserve to be taken seriously. It means that labeling you with a mental diagnosis and treating you for it is unlikely to have a net benefit for you, or to the mental health system.
I always use the example of a patient who hears voices. My first questions to this patient would be whether or not he finds this distressing, and whether or not his hearing of voices has caused him interpersonal problems with other people (causing him to lose friends or be unable to pay attention in class, for example). I would then probe for other odd mentations and behaviors, since hearing voices is a common symptom of several types of mental illness. But if the voices didn't seem to be causing the patient any problems, and he didn't exhibit any other mentations suggestive of a diagnosable disease, I'd withhold treatment. Granted I'd warn the patient to come back and see me if either of these conditions changed. I'd document the hearing of voices and the patient education on when to come back and see me. But the point is, hearing voices in and of itself is not enough to diagnose or treat anything. Just because I don't hear voices, and nor do most of my patients, does not put me in a position to deem that mental action something worth fixing. "If you hear voices, you're crazy" would be a value judgement on my part, and just bad medicine. I'm not saying this kind of thing doesn't happen all the time. But it's wrong.
Does society care about the mentally ill? Again, ask yourself, does society care about the physically disabled? In both cases, it's not a black and white thing, it's a matter of priorities. Affluent, stable societies can afford to turn their attention to making sure all public buildings have ramps, and all public school districts have a mental health professional on the payroll. Societies faced with shortages of basic resources or capable leadership, on the other hand, tend to take a more sink-or-swim attitude toward members who can't pull their own weight, even through no fault of their own. This of course opens the door to deeming all sorts of people expendable. Therefore, I say any healthy society works constantly to ensure the basic safety, security, and material wellbeing of all of its members, so that attention can be paid to helping less able members find a niche.