Empathy vs. Narcissism

Empathy is when we recognize the feelings and different point of view of another, and are able to "put ourselves in their shoes". It's a brain function of humans and other animals as well; it's one of the ways we seem to automatically "sense" what another person is feeling, or what they want or need. We often actually experience pain when we see another injured, either physically or otherwise.
It's a brain function process, just like other brain processes, that help us navigate and understand our world.

Those who seem to have little or no empathy function can have trouble understanding "where another is coming from", or how they feel, or why they feel that way. They may not be able to tell the difference between expressions of different emotions coming from others, and may mix them up.
For example they may think a person who is afraid and anxious is actually expressing anger and domination.
They may think a person who is expressing sadness is acting "stuck up" or being self-centered.
They may see a person's body language who is expressing joy and celebration as being aggressive, and they may think that a person who is being condescending and arrogant is exuding confidence and intelligence.

A person who has trouble with empathy can react with fear or anger, even rage, toward a person who is emotionally or physically experiencing pain, because they are completely misinterpreting the person's expression as aggressive, controlling, or demanding toward them.

Having an apparent lack of empathy can come from many things, from traumatic experiences, a hostile upbringing or community, to brain damage from trauma or certain drugs, to simply being born with a lack of neurons (brain tissue) that allow one to experience empathy.

Empathy is an actual brain function (not something that's "moralistic" or "new age", it's an evolved ability)  that helps us function in the world, and helps us understand what is happening around us. It's not some kind of "virtue" or special power, nor does it naturally occur in only one race or sex but not another. The propensity for empathy is individual, like math ability. It can be increased with guidance, learning, and understanding more about it. There are ways to learn about empathy and increase one's own ability and understanding, which can open doors for us both personally and professionally, in family, relationships, and in the business or academic world.

The first step to empathy is realizing that
EMPATHY IS NOT SOMETHING WE GIVE OR PROJECT, it's something we are able to DO.
Some people are naturally better at it than others. Some people are very good at it, some not.(Like math).
The better a person seems to be at empathy, the more they understand why it's important, and usually the less skilled a person is at empathizing, the less they understand why it's important, and will often see it as a waste of time to learn more about it. (Just like math.)
Further, there are many who think they are experts already, and have arrogance about their ability to empathize so they refuse to learn more or self-reflect, which actually hinders their ability. (Still, just like math.)

A person who is good at empathy would be able to understand cause and effect in another person's life and situation, to see situations and people with clarity, and is much less likely to make assumptions, envy, be aggressive, act with hostility, and project their own motives and negative judgments onto others. Empathy or the lack thereof can mean the difference between talking and listening with a person vs. countering, rejecting, or raging. It can mean the difference between standing up for someone and 'throwing them under the bus'.
It is the difference between respecting another's dignity and reputation, and slandering or humiliating them.
It can mean the difference between a person receiving the assistance and help they need vs. being treated with dismissal, invalidation, rejection or hostility. It often means the difference between a parent who loves, guides, protects, teaches and cares, and a parent who does not, who neglects or attacks their own child.
It is often what makes or breaks all kinds of human relationships, be they personal, professional, or business.

Ironically, everyone seems to expect empathy and understanding from others, regardless of their own ability.
So it's not always that they don't know what it is, it's just that they either don't understand how to do it, or are unable to do it, or don't want to do it because they don't like things that seem like work (again, like math). 

Here is one article about it, with exercises included:

The Empathy Workout by Martha Beck 

Click any text for the full article:

"Caveat Empathor

Many people, especially those of us who’ve had a little bit of therapy, fall into an emotional trap Buddhists call “idiot compassion.” At first glance, this looks like empathy, but it’s actually projection. It encourages us to condone harmful behavior by assuming that the perpetrator is acting out of pain and helplessness. 
“I know he’s just a hurting little boy inside,” says Jeanie, whose boyfriend, Hank, has just beaten the living tar out of her for the umpteenth time. “He’s so sensitive. His mama abandoned him. He even cries when he talks about it.” Because Jeanie herself would become violent only in the grip of intolerable torment, she thinks she understands Hank’s motivations—and so she excuses his behavior. Real empathy is not based on this kind of projection but on close observation. If she were a true empath, Jeanie would notice that Hank, while “so sensitive” to his own misery, never notices others’ distress. 
When Jeanie understands that no one who cares for her could act as he acts, she’ll drop the idiot compassion and get the hell out of Dodge. At that point, she’ll realize that real empathy doesn’t put us in harm’s way. It protects us. That’s just another reason to implement one of the following exercises:

Exercise 1: Learning to Listen 

If you want to feel that you belong in the world, a family, or any relationship, you must tell your story. But if you want to see into the hearts of other beings, your first task is to hear their stories. Many people are gifted storytellers. Only the empathic are true storyhearers.

Anthropophobia: the fear of people