Empathy is defined as the ability to understand (cognitive empathy) and share (effective empathy) the feelings of other people.
This may be somewhat complicated by alexithymia, the ability to feel and describe your own emotions.
It may also be confounded by the different ways Autistic and allistic people tend to relate to emotions in terms of communication style. For example, if someone describes a plight of theirs, we might offer a similar story of our own to show how much we can relate to it. Allistic people will likely misinterpret this as an attempt to one-up them with a disregard for their feelings, rather than a sign of camaraderie.
Many Autistic people have what allistic people regard as "too much" empathy, as we'll be ethical even when no-one's watching.
Some of us also have a tendency to ascribe emotional states even to inanimate objects, quite possibly explaining the prevalence of plushie collections within the community.
It's possible that allistic people generally only have empathy for people they know and are familiar with, while not having empathy for strangers. Bear in mind that social circles are somewhat neurologically limited to about 150 people, suggesting that most people do not have empathy for most other people, only for their tight-knit group of friends and family. (Verification is definitely needed for this claim, even if it would explain a lot.) If this is true, it's certainly hard for those of us who apparently have hyper-empathy to wrap our heads around.