Ásatrú is an Old Norse compound derived from Áss, which is the Old Norse singular referring to the Ases or "Anses", and trú, literally "belief" or "faith". Thus, Ásatrú is the "Belief (or faith) in the Gods". The term is the Old Norse/Icelandic translation of Asetro, a neologism coined in the context of 19th century romantic nationalism, used by Edvard Grieg in his 1870 opera Olaf Trygvason. Ásatrúar, sometimes used as an adjective in English, is properly the genitive of Ásatrú.
Modern Scandinavian forms of the term, Norwegian Åsatru, Swedish Asatro, Danish Asetro, were introduced in Neopaganism in Scandinavia in the 1990s.
In Germany, the terms Asatru and Odinism were borrowed from the Anglosphere in the 1990s, with a chapter of Odinic Rite formed in 1995 and the Eldaring as a partner organization of The Troth formed in 2000. The German Eldaring takes Asatru as a synonym of Germanic Neopaganism in general, following usage by The Troth. Eldaring is the only pagan organization at the national level in Germany self-described as Asatru. Other institutions prefer German words such as Alte Sitte ("Old Custom") or Germanisches Heidentum (simply "Germanic Heathenry").
Thus today the term "Ásatrú" can be seen used as both a name for a specific undercurrent within Norse Neopaganism worshiping primarly the Ase gods; as a synonym for "Norse Neopaganism" (especially in the American Asatru movement); or less and less commonly as a synonym for the whole Greater Heathenry.
There are three national organizations of Nordic Paganism in the United States, Ásatrú Alliance, Ásatrú Folk Assembly and The Troth, besides numerous smaller or regional associations. Historically, the main dispute between these has generally centered on the interpretation of "Nordic heritage" as either something cultural, or as something genetic or racial ("metagenetic"). In the internal discourse within American Ásatrú, this cultural/racial divide has long been known as "universalist" vs. "folkish" Asatru. The Troth takes the "universalist" position, claiming Asatru as a synonym for "Northern European Heathenry" taken to comprise "many variations, names, and practices, including Theodism, Irminism, Odinism, and Anglo-Saxon Heathenry".
Source: Wikipedia, "Germanic Neopaganism", available under the CC-BY-SA License.