Several real-world mental disorders are associated with a Lack of Empathy. The real-world situation informs fiction but neither really mirrors the other.
"Empathy" is described as the ability to comprehend the signals other sentient beings (mostly humans, but animals too) use to communicate emotion and being able to understand these emotions from their perspective, which allows a person to experience sympathy and compassion. Think about all the Emotion Tropes we use to identify and/or display emotion—Puppy-Dog Eyes, Manly Tears, Oh Crap, Something Else Also Rises; (mostly) universal things that make sense even if you're watching a foreign film with no subtitles. Someone who lacks empathy lives in a world completely devoid of these signals. Accordingly, such a person is a Fish out of Water whenever they are around people... which, of course, is much of the time. They can also come across as a Jerkass who seems to think It's All About Them.
There are two main different types of empathy: Cognitive Empathy and Affective Empathy.
Lack of Cognitive Empathy
Cognitive empathy is the ability to recognize and understand another's emotional state.
People who lack cognitive empathy will be unable to read the emotions others are feeling and/or after knowing how they're feeling, will fail to understand why they're feeling that way, not being able to put themselves in their shoes. But once they do, they are capable of emotionally empathizing with them.
Those with Asperger's Syndrome and other autism spectrum disorders have difficulty consciously recognizing nonverbal language, but usually have compassionate empathy. Meaning that they share in others' emotions as easily, if not moreso, than average. In particular, feeling pain.
Another problem is that even when they share in others' emotions, they are still unsure on how they are supposed to react and have problems coming up with an adequate response.
People with Borderline personality disorder have very intense emotions, they feel them more easily, deeply, and longer than other people. This intensity can cloud their perceptions and make them mistake the emotions, thoughts, and intentions of others.
While people with this disorder are very good at recognizing other people's emotional expressions, their perception of them can be more extreme than it really is and often tend to take said emotions personally, thinking they're directed at them and can be so overwhelmed by their own feelings that they fail to give an appropriate emotional response.
Bipolar disorder is a mood disorder in which people have mood swings, but unlike with what is popularly assumed, they go from "normal" to mania to depression with each mood lasting for extended periods of time, commonly weeks or months.
Similar to BPD above, they identify emotions well, but their emotional state can make them misinterpret others' mental states and sometimes their emotional response is exaggerated and expressed in a dysfunctional overreaction also called "hyper empathy".
Lack of Affective Empathy
Affective empathy (or emotional empathy) is the ability to share the emotional experiences of others, being affected by them and being able to respond with an appropriate emotion.
People who lack affective empathy can infer and understand what others are feeling; they can even know what emotion it should elicit in them in response, but they just don't feel it, which makes them incapable of relating to other people, although depending on the disorder, they will be able to imitate and pretend to empathize.
The Lack of Empathy trope refers to this type of empathy.
Of course, we have to mention Antisocial personality disorder, more popularly known in the past as The Sociopath.
In general, sociopaths experience the world so differently as to be effectively unperceivable by those with any grasp of empathy. They feel anger, but it tends to be short-lived, often lasting only seconds, though the anger is often explosive and uncontrolled. They are very egocentric and callous, they do not appreciate the consequences of their actions, and they tend to not feel guilt or remorse. They can be, however, very, very good at reading other people's emotions and imitating them; this enables them to be highly sociable, and some people live normally with undiagnosed ASPD for years and by extension all their lives, appearing to be a law-abiding and healthy member of society simply by mimicking the law-abiding behaviour of those around them (although they do not grasp why this is the accepted social standard).
ASPD does not necessarily mean the sufferer has high intelligence, but this assumption was probably gained from their Manipulative Bastard abilities. It is considered almost untreatable by those in the psychiatric profession, as many of them see their doctor as just another person to be manipulated.
While sociopaths have a degree of overlap with narcissists and the like, the difference is that the Sociopath won't take it personally if you humiliate him and kick him out. The only reason he'll stay in somebody's life is because that person is gullible (or he classes that person as gullible) and there's no need to reinvent the wheel. For example, while narcissists usually see themselves above other people, when in doubt they want those very "lowly others" to take care of their needs and wants. Sociopaths will simply take what they need or want. Narcissists are still bound by emotional, moral, and conscientious weaknesses, such as love or guilt, and when cornered, are far more easy to be punished. Sociopaths can easily disregard such weaknesses on the go, and simple punishment usually does not work on them.
Psychopathy and Sociopathy
You know how we said people ASPD and sociopathy were the same thing? That's kinda only trueFrom a Certain Point of View. Publicly, sociopaths, psychopaths, and people with ASPD are thought of as the same thing. Officially, there is a distinction, albeit a very blurry and not entirely consistent one.
Psychopathy is not a diagnosis that can be found in the DSM, which instead recognizes ASPD. However, it is an important concept in forensic psychology as a means of risk assessment, and is used to inform legal decisions regarding criminal offenders' sentencing or parole hearings. Much like "insanity", "psychopathy" has more legal connotations than clinical ones. It's also this guy specifically that's described in The Sociopath trope page.
ASPD is the DSM's answer to psychopathy and was established out of concern that the diagnostic criteria of the psychopathic personality was too subjective. Because personality characteristics cannot be assessed objectively, ASPD diagnostic criteria disregards most of them and instead is based predominantly on the presence of a consistent, long-lasting pattern of antisocial behavior. As chronic antisocial behavior is not exclusive to a single disposition, this means that the diagnosis of ASPD covers a significantly larger population than that of a psychopathic personality — most psychopaths fit the criteria for ASPD, but most people with ASPD would not be considered psychopaths according to the PLC-R.
The difference between psychopathy and sociopathy is fuzzier. The term was introduced in 1909, reflecting the new suggestion that psychopathy was largely a product of social factors. It eventually came to be used to describe "secondary" psychopaths — people who shared many characteristics with "primary" psychopaths, but were a product of severe social maladjustment and were capable of experiencing emotions of depression or fear. However, there is no official diagnosis of sociopathy/secondary psychopathy, and its distinction from psychopathy is more a matter of preference than fact.
A lack of empathy is not uncommon in serial killers, who are commonly diagnosed with ASPD. That being said, your average ASPD sufferer is more than likely to be a petty criminal, or simply a law-abiding Jerkass. Also, it is important to note that lack of empathy does not make you a serial killer, or even a Jerkass; they simply make the person more noticeable in advance, if they should happen to act out in some way. It does make it more likely that the person in question will do something mean or exceedingly callous (in other words, act like a Jerkass), but this isn't automatic; it's just a consequence of failing to consider other people's feelings.
Despite the criminal justice system's treatment of premeditated murder in favor of crimes of passion, it is important to note that all people without emotions are not serial killers (they might in fact be more likely to be schizoid (see below), which in fact are for the most part harmless, and are more in danger of being alone their whole lives than going on a murdering spree). The average person avoids the unshaven type, but a serial murderer is actually more likely to be handsome, charming, and seemingly normal.
Narcissists can identify people's emotions and use this information to manipulate them to get what they want and need, many times considering other people as just tools, callously ignoring if they're hurting them in the process. In short, they need people but don't care about them.
People with Schizoid personality disorder can also lack empathy. People with the disorder have flattened affect, meaning that they don't feel much of anything. They can go from being pretty much emotionless to experience very shallow emotions that tend to disappear quickly.
Schizoids are perfectly capable of reading other people (if bothered) and even able to understand their perspective in a rational way, but is hard for them to feel for others when they don't have much feelings to begin with, which makes them extremely detached from others, including family. Adding to this, they also have an aversion and/or inability to express the feelings they do have, so even when they do care, they will still appear cold and indifferent.
It's important to note that while schizoids don't have much emotions and don't care about others, they're not dangerous and can have a strong moral code. Unfortunately, quiet, aloof loners and asocial people in general are often seen with suspicion and accused of being psychopaths or time bombs ready to explode when in reality schizoids (who are the most extreme version of loners) are very cool-headed and also lack motivation, so going out of their way to harm others would be seen as unnecessary work that wouldn't reward them in any way. In fact, they care so little about everything, including themselves, that they're more prone to be a victim than a predator. Also important is not to confuse schizoid with schizophrenia, even if they sound similar, and while SPD is considered to be in the schizophrenia spectrum because they share some of the characteristics mentioned above, schizoids don't suffer from hallucinations, delusions, and the like (see below).
Those with Histrionic personality disorder can be unempathetic, stemming from being so focused on a group of people they lose sight of the individuals making up that group. Unlike The Sociopath, they are more embracing of social groups, but will move on to other people, with no regard to what effect this has, if they don't find enough stimulation there.
They also may express empathy or affection, but the actual level of emotional connection quite often ends at the surface. Someone with HPD can mimic, or even exaggerate, an appropriate emotional response, but his or her selfish nature often prevents a true bond with others.
Schizophrenia is a severe mental disorder where people have difficulty distinguishing between what's real and what's not. Obviously, everyone associates this disorder with the always popular symptoms of Hallucinations, Hearing Voices, delusions, paranoia, and disorganized thinking and speech. Less known, though, are other emotional symptoms such as flat expression, lack of emotions, motivation, interest in relationships, and empathy. Like schizoids, these characteristics make them incapable to relate but unlike SPD, it's made even harder by the fact that they also lack cognitive empathy, having severe impairments in social cognitive tasks.
And by the way, schizophrenia is not the same thing as what is popularly known as Multiple Personality Disorder (now called Dissociative Identity Disorder). Schizophrenics do not have different identities or personalities within themselves, they just hear and see things that aren't there. That said, it's entirely possible for a person to both be schizophrenic and have DID.