Emotional empathy includes the ability to understand and respond appropriately to the emotions of others. Such emotional understanding can cause someone to worry about the well-being of others, or it can lead to feelings of personal distress.
Physical empathy includes specific physical reactions depending on what someone else is experiencing. People sometimes physically experience what others are feeling. For example, seeing someone else feel embarrassed can cause your face to blush or your stomach to feel sick.
Cognitive empathy includes the mental state of others and the ability to understand what they are thinking in some circumstances. This is related to what psychologists call the theory of mind, or thinking about what others are thinking.
Sympathy and compassion are related to empathy, but there are significant differences. Compassion and empathy are often thought of as having a more passive connection, but empathy generally involves a much more aggressive attempt to understand others.