As with most Neo-Pagan religions, Asatru posits a belief in magic and the spiritual realm. However, people must remember that the bedrock of Asatru is faith in the Gods, and magic is but a part of our customs and folklore, not a substitute for faith or something separate from it. Practicing magic, even magic of a Northern type, does not make one Asatru, nor is the practice of magic a requirement to be an Asatruar or to perform rituals in honor of our Gods.
The most common type of magic found in the Asatru tradition is that of the runes. The runes are a magical alphabet which in various forms was found throughout the Germanic world. The most common form used in Asatru today is the "Elder Futhark" (runic alphabets are called futharks, a word constructed from the first 6 runes) of 24 runes which is believed to be an older and more true form than the later versions such as the Anglo-Saxon set of 33 runes.
People are most familiar with the use of runes for divinatory purposes, and they are indeed used for this purpose. Asatru believes that there are forces, shaped by our past and the history of the world, that affect the world and the way the future comes to be. We believe that the forces of Wyrd and Orlog (without a dissertation to explain them fully, both words translate roughly to "fate") can be examined and to some extent tell us what is going to happen. On the other hand, we do not believe in predestination. Future events are shaped by our actions, and we can change them. If we change our actions, we change the future. So the runes are not a perfect prediction of what will occur because the future is in flux. They are, however, an important tool for exactly the same reason.
A common way to read the runes is to pull forth three runes representing the past, present, and future. All of these are important, because only in looking at the past and present can we understand a prediction of what will occur in the future. Another simple way to read them is to pull forth a handful (4-6) and drop them onto a cloth, examining not only the runes, but how they fall in relation to each other.
However, divination is but a small part of runic magic. The runes are important and powerful symbols that represent the very forces that hold the nine worlds together, and they make very powerful meditational symbols.
The runes are also useful in active magic. The most common way to use them in this manner is to carve a "bind rune" or a symbol made up of more than one rune, all of which together are intended to produce an effect. The most common of these would be a rune carved on a single line with one rune pointing to the left and the other to the right. However, the more complex a rune is, the more powerful it can become.
No overview of runic magic would be complete without mentioning "galdr" or chant magic. The simplest form of this is "rune galdr" or the simple chanting and vibrating" of the sounds of the runes in order to invoke their powers.
For more information on runes, consult the books recommended in the appendix.
Another important type of magic is called seidhr, which seems to have been a "shamanic" tradition within ancient Asatru. Sometimes, these practices are called spawork. It has been suggested that these were similiar practices, but that the first term involved magics that were considered disreputable, and the second term, magics that were considered reputable. Some modern practioners make this distinction, some do not. This type of magic involves going into a trance, speaking to spirits, and journeying to the other worlds. One might consult the spirits of nature, the Disir, or the ancestors. Unfortunately little information is left to us. We know that Freya was a skilled practitioner of seidhr and that she taught it to Odin. It was considered to be a woman's magic, and Odin is taunted about it by Loki. Although today most persons exploring seidhr are women, there is no such prejudice against men interested in it.
In what records we do have, the trance of the seidhrwoman was created through another person singing songs or chanting while the seidhrwoman was elevated on a platform. We don't know much else about the practice. However, around the world shamanic techniques are remarkably similar, and the main difference seems to be the cultural context, which provides a map to interpreting the otherworlds. The best approach might to be explore some of the material on core shamanism (shamanism studied outside of a specific cultural context), and then apply that to what little we do know.