I found the following description from Building Moral Intelligence very helpful:
The first year of life - global empathy. The child cannot clearly distinguish between himself and his world, so he is unclear as to who is experiencing distress and interprets it as his own. The six-month-old baby heard another baby cry and began to cry too.
Beginning around age 1 - egocentric empathy. The child's reaction to others in distress begin to slowly change. She now understands that another person's discomfort is not her own. The two-year-old saw her mommy crying, so she sat next to her and softly patted her arm.
Early preschool years - emotional empathy. Around the age of two or three, the child begins to develop role-taking capabilities. He recognizes that someone's feelings may be different from his, is better able to decipher the source of another person's distress, and finds simple ways to offer comfort or show support. You look sad. Your car broke. You can use this one.
Cognitive empathy - early school years, beginning around age 6. The child can now see things from another person's perspective, so there is a noticeable increase in her efforts to support and comfort those in need. The ability to use language to comfort others also substantially increases. That old woman looks like she needs help getting into the elevator. I will hold the elevator door open, so she can walk in safely.
Abstract empathy - late childhood: ages 10 to 12. The child can now extend empathy beyond those he personally knows or can directly observe to include people he may have never met. The people in Africa look so hungry. If I sent some of my allowance each week, it might make them feel better.
It made me realize that maybe I push Anna too hard by trying to get her to comfort her playmates when they are in distress. She is still in age-appropriate stage of emotional empathy, when she tries to sort out the feelings of this other person. She is pretty observant and very sensitive to moods. Even when she looks at the pictures, she asks sometimes, Why is he sad? or Why is she worried? She always asks us why is he/she crying? when she hears someone in distress. But strong emotions of others scare her. She is raised in a quiet home, where everyone is polite. My husband and I rarely argue in front of her (and we rarely argue in general). A couple of times we did it really freaked her out, and she kept asking for weeks, What did papa say? Why were you loud? She is able and willing to offer comfort to us - when one of us doesn't feel well, she will come and cuddle and give many hugs and kisses and tell us to feel better. But she is not willing to extend herself to playmates, because she doesn't have any real friends yet. I now feel more comfortable that she is simply too young and empathy for playmates will come with age.