Try explaining empathy to someone who has limited understanding of empathy. I find this activity a challenge, having attempted it many times, so I am interested when I see new attempts at explanation. Currently I am reading Buddha's Brain by Rick Hanson. The book is about Mindfulness and neuroscience, and in it he describes empathy in a practical way.
Beginning on page 125, he states that "empathy [is] the capacity to sense the inner state of another person...". He further breaks empathy into our ability to "simulate" (I like this word) another person's actions, emotions, and thoughts.
Regarding actions, similar parts of our brain are activated both when we do an action or perceive someone doing that action. This is the effect of "mirror neurons", which are actually networks of neurons. We also experience a similar neural process when we both have emotions and see someone having the same emotions. Interestingly, he stated that greater awareness of your own emotions leads to greater ability to read the emotions of others. Finally, Theory of Mind (ToM), one's ability to "think about the inner workings of another person" is an ability that is likely exclusive to humans (my opinion) and largely the product of the prefrontal cortex.
Hanson goes on to suggest that these three "systems" work together. For instance, one can gather data (i.e., perceive and notice) about the actions and emotions of someone else, put it through the ToM processor, and create a model that you test on yourself (e.g., imagining yourself acting and feeling a certain way). Essentially, you can simulate and thus experience the experience of another person. This is called empathy.
The application for empathy is enormous and outside the bounds of this post, but I thought the explanation of the components of the act of empathy were interesting. I would also like to point out that much of the act of empathy, according to Hanson, appears to be noticing or guessing about the actions of others, a skill individuals on the Spectrum are not very good at (due largely to executive functioning deficits and also possibly lower affiliative drive). This explains, to me, why empathy is present, but often under developed in Spectrum individuals.