Do unto others as you would have them do to you. This is not only the Golden Rule, but one of the ultimate statements of pure empathy. It is a simple lesson in how to relate to others as well as how to get along. It is not merely advice of action but of thought.
Allow yourself to take your own experiences and imagine someone else in your place. Now reverse this exercise and try to imagine what someone else is going through. Perhaps your neighbor has had a death in the family recently. How do you react? How do you feel? Especially if perhaps somehow you have not yet experienced such a deep loss yourself. Maybe you've experienced the 'lesser' grief of losing a beloved pet or a dear friend when they moved away. Loss is loss, no matter what shape or size. Allow yourself for just a moment to feel again as you did when that event first occurred. What did you want most at that time besides the return of your loved one?
Empathy can also be joyful. Your neighbor just won the lottery and you're feeling jealous instead of happy for them. How do you correct these negative feelings? Imagine your best day. Most people have had at least one good or even beautiful moment somewhere along the way. Now look again at the smile of your neighbor and remember those good feelings of your own so that you can celebrate their current triumph instead of being jealous that someone has it better than you.
We all have experiences. We all have feelings(with a few odd exceptions). We are all capable of understanding each other by taking what we know and using that knowledge to sympathize with others. We are all capable of empathy.