There is no doubt that some children are wired up to be more naturally empathetic than others. For example, the children I teach tend to be terrible at empathising because in cases of very poor attachment, developing empathy is quite low down on the list of 'essential for survival'. The good news though is that most children can be taught to empathise. You can learn empathy without being 'naturally' sensitive or tuned into emotions.
There are different opportunities for teaching empathy. Here are a few ideas:
When watching TV programmes or reading stories you can ask your child to imagine how different characters might be feeling.
Help your child to understand that in order to work out what another person is feeling, you need to listen to them carefully, not just the words but their tone of voice also and look at the expression on their face. There are usually visual and auditory clues to how another person is feeling.
When looking at photos of faces, you can ask your child what they think the person might be feeling.
You can offer an emotion e.g. embarrassed, excited etc and ask your child to think of a situation where a person might feel that emotion.
When your child tells you about any fall-outs they might have had, ask them to try and imagine what the person they have fallen out with might be feeling.
Ask your child to imagine how they would make someone feel if... .and give a selection of scenarios e.g. you gave them a compliment, you pushed in front of them in a queue, you told a lie about someone etc. This helps your child put themselves in the shoes of others.
As a role-model try and express how you feel in different situations so that you child can link emotions to likely causes.
When children develop empathy, they become far better at dealing with emotionally difficult situations (e.g. a friendship conflict) as simply being able to speculate how others might be feeling can make conflict resolution more likely as ultimately empathy can make a child more able to forgive.
There is a tool in neuro-linguistic programming (NLP) that uses empathy to resolve personal conflicts. NLP recognises that some people tend to predominantly look at things from their own perspective only while a smaller number tend to predominately automatically look at others' perspective. (This latter group can be very giving but can suffer from never putting themselves first). NLP recognises that to sort out a conflict, all parties need to be able to see each others' perspective. The tool goes further and suggests that those involved in a conflict imagine a third person (perhaps a wise grandmother or wizard!) who wasn't involved in the conflict and consider what they might say about the situation. This 'empathy blast' can help resolve situations quite quickly.