"Asatru" is the name of a pre-Christian religion of Northern Europe, specifically based on Norse/Germanic ("Viking") mythology, It has numerous gods, such as Tyr, Odin, Thor, and Freya, among many others, major and minor. The most accessible current book on the mythology underlying this religion is Kevin Crossley-Holland's "The Norse Myths".
The formal name "Asatru" (pronounced OW-sa-troo) is a fairly recent invention of the 19th Century: if the Viking Age people had an explicit name for their religion, that name has been lost. The casual name for this religion is heathenry. Whether Asatru can or should be classed as neo-pagan is hotly debated, as it reconstructs a religion that is known beyond question to have historical roots. Many heathens distance themselves from practitioners of more popular modern pagan religions such as Wicca.
Asatru is a "hard-polytheistic" religion. Its many gods are seen as distinct, not as specifics of some larger generalization. Furthermore, the gods of other religions are acknowledged, but not worshipped. Heathens are quite willing to respect other religions, on the condition that such respect goes both ways. Proselytizing is frowned upon, but discussion is not.
Asatru has some similarities to the Japanese Shinto religion: respect for nature and its forces, and veneration of ancestors. A list of explicit virtues is promoted, and the swearing of oaths (not to be confused with foul language) is taken very seriously.