Is there a word for the ability to sense something someone else is feeling? Kind of like the opposite of autism?
Empathy: The feeling that you understand and share another person's experiences and emotions : the ability to share someone else's feeling
The word "empathy" specifically describes that ability.
Empathy (n.): the ability to share someone else’s feelings or experiences by imagining what it would be like to be in that person’s situation. Cambridge English dictionary
Are you thinking of telepathy - the supposed communication of thoughts or ideas by means other than the known senses. (Oxford Dictionaries Online).
Even stronger than empathy, there is Mirror-touch synesthesia. More of a condition than a sense though.
Wikipedia describes it as
a condition which causes individuals to experience the same sensation (such as touch) that another person feels. For example, if someone with this condition were to observe someone touching their cheek, they would feel the same sensation on their own cheek.
a supposed intuitive faculty giving awareness not explicable in terms of normal perception.
You can also say someone is especially perceptive, or sensitive. In some contexts clairvoyant could work too, though this is more about seeing events in the future.
Also, a mind-reader:
the ability to discern the thoughts of others without the normal means of communication, especially by means of a preternatural power.