Paganism and Heathenry are a general classification that encompasses the religious beliefs of pre-Christian Europe. The main difference between the two terms is that Heathenry focuses on Norse and Germanic beliefs, whereas Paganism is more general to all of Europe, and also includes Greek or Roman beliefs. In practice and popular culture, however, both terms are generally inter-changeable.
Asatru, on the other hand, is a subgroup of Paganism. Asatru is to Paganism what the Westboro Baptist Church is to Christianity. Asatru is a very Christianized and bastardized subgroup of Paganism, with very little connection to actual Norse culture and beliefs. The dogma of Asatru is actually generally in complete contradiction with Heathenry concepts.
Forn Siðr is another subgroup of Heathenry, and is the closest thing to actual Norse (not Germanic) culture and beliefs. Forn Siðr actually translates to "Old Customs". We may sometimes describe Forn Siðr as "Norse Paganism" for those who are not familiar with the term or the culture.
Forn Siðr is not a religion or faith to us. It is our truth, our ancestors, our culture, our identity, our knowledge, our masculinity, our spirit. Forn Siðr is what makes us men and our own man, taking our own decisions, and taking our own responsibilities. Forn Siðr is a warrior belief system.
We do not pray to the Norse Æsir, Ásynjur, Vanir, Jötnar, Aðrir, Kindir, or Kynja (Old Norse terms do not translate to "gods"), nor do we expect them to tell us what to do or how to behave. They are our ancestors, they are ideals we strive to achieve, they are ourselves, they are us.
A fundamental aspect of Forn Siðr is that it has no dogma. No centralized structure, organization, or power. No one to tell you what to think or believe, how to behave, or how to live your life.