Negative liberty is the absence of the initiation of force. Free speech is an example of negative liberty. The right to free association is another example of negative liberty. You have the right to associate with whom you see fit, but it doesn t force others to associate with you. You can start a club, but you can t force people to join.
Positive liberty is the freedom to something in order to exercise a right.
The right to education is an example of positive liberty. In order to exercise that right schools have to be paid for and for there to be universal education, tax dollars will have to be used. The right to health care is another example. Hospitals, doctors, nurses, drugs, and medical equipment aren t cheap, and for everyone to have these readily available, the money has to come from somewhere. The fact is positive liberty is not really liberty at all.