The quick ‘n’ dirty:
Heathenry or Heathenism is an umbrella term applied to the revived/reconstructed religions of the pre-Christian Scandinavians, Anglo-Saxons, and continental Germans. Broadly speaking, the term “Heathenry” can be applied to Germanic neopaganisms.
Ásatrú is the term applied to the modern Icelandic/Norse-focused branch of Heathenry. As a modern religion, Ásatrú was founded first in Iceland in 1972 under the Ásatrúarfelagidh. Other organisations have since popped up in other countries, many in other branches of Heathenry. Ásatrú is sometimes used to refer to Heathenry in general but is more accurately applied to a Scandinavian and especially Icelandic/Old Norse focus.
Heathenry can differ widely in its various forms, but one commonality is that its branches are generally polytheistic, and follow a general pantheon of gods (with small and large regional differences). Broadly, we’re talking about gods like Odin and Thor and Frigga, who are known by different names in different places. In addition to gods, Heathenry often acknowledges spirits of the land and ancestors. Practitioners may also include folkloric influences in their practices.
Heathenry is informed by the literary and archaeological sources available to us about the pre-Christian heathens. This ranges from poetry to runic inscriptions, image carvings to burial goods, sagas to statuettes. All of these vary, again, depending on region, thus allowing for the different branches of Heathenry.
Heathenry is not just about honouring gods and spirits, it is about a philosophy of life and an understanding of culture.
There is nothing inherent in Heathenry or any of the sources available to us that implies a racist ideology. Any racist, neo-Nazi, nationalistic rhetoric you may see slapped onto Heathenry have come from somewhere else. Heathenry is not about race or nationalism. Anyone may practice it. It is an open group of religions.