My understanding is that empathy means to feel what another person is feeling while sympathy is the ability to understand what another is feeling.
Here are two definitions from the web:
“Both empathy and sympathy are feelings concerning other people. Sympathy is literally ‘feeling with’ – compassion for or commiseration with another person. Empathy, by contrast, is literally ‘feeling into’ – the ability to project one’s personality into another person and more fully understand that person.” 
“To express sympathy is to make it known that you are aware of another’s distress and that you have compassion for them.
To express empathy takes things a step further by not only expressing compassion but also showing a deeper level of understanding by entering into the other person’s experience.” 
Definitions 1 and 2 say the same thing about sympathy: an awareness and compassion for another’s feeling. I understand this and do not disagree. I would revise my own definition above, to say sympathy is understanding with compassion.
Definition 1 states that empathy is the ability to project one’s personality on another to understand that person better;
Definition 2 states virtually the same thing: empathy is sympathy plus entry into another’s experience.
Is empathy even possible?
Is seems impossible to filter the world through another’s personality. One’s personality is the sum total of events that happen to that person and only that person. To project one’s personality onto another seems equally absurd; in that case, there is high likelihood that prior experiences and reactions to events will be slightly, if not totally different.
The best-case scenario is to share an extensive experience with the other person, and share thoughts about them. Under these conditions, there is a higher likelihood that at least some empathy is possible.
There are empathetic people, but how?
I know there are many empathetic people in the world. Is it a matter of guessing how another person is feeling based on how you would feel under similar conditions? Putting yourself in the place of the other person? Is it as simple as that?
Does anyone have a better definition of empathy?
Do you think my mother is sympathizing or empathizing with my brother Mike in the pictures below?
My brother Michael has never spoken, he is very low functioning and autistic. Is it possible to have empathy with him or is sympathy the only option?