Asatru is a belief in the Germanic gods. It is not a structured religion in the sense of other mainstream religions (e.g. here's a book now follow it). The knowledge for the practise of it comes from the Edda's, saga's & other accounts of the beliefs (unfortunately no real accounts from those who followed them), as well as coming from archeology and linguistics.
It is often said to be the "religion with homework". I am a deconstructionist Heathen which means I'm more interested in reconstructing the worldview of the Germanic people (for me with a greater emphasis on Anglo-Saxon Heathenry).
It is the Pagan/Pre-Christian European religion that I have practiced for over 15 years. Asatru is a reasonably accurate reconstruction of the beliefs of the Viking-Age Scandinavians (except for the Saami/Lapps and Finns). There are other Germanic Pagan beliefs. Odinist is another name for more generic Germanic Paganism (could be Anglo-Saxon, German or Dutch, as well as other Germanic tribes that never founded their own countries or that didn't survive into the present). Odinism can mean more than one thing too; like someone specifically dedicated to Odin.