The astral realm is a phrase used loosely to describe the quite possibly infinite levels of reality and perception, most specifically the reflection of our physical realm.  Yet the most commonly experienced level of reality, beyond the physical, it the realm of dreams. It is a place where sight, sound, and feeling are perceived as “real” whilst the dreamer is inside.  While the multiverse is considered by most magick users to be made up of an immeasurable number and variety, there are several places generally considered to be the realms most common to the experience of a magus.  These are the astral, the infernal, the dreamscape, the celestial, and the realm of faerie.
The astral realm is the spiritual reflection of the physical realm. While much of the realm is identical to the physical realm, the energy resonance of places is tangible in the astral. The infernal realm is a distillation of the negativity and more evil or demonic elements and denizens of reality, many others refer to this realm as the lower astral.  The celestial is a realm of ideals, positive energy, and is the repository for most of humanities higher spiritual aspirations.  The realm of faerie is the patchwork world of folklore, myth, and fantasy.  The realm of dreams is less its own reality and more a staging area for entrance into one of the above realms.
Some call the astral the spiritworld, others simply refer to it at the “other place”. To make the attempt to classify levels of existence that defy classification would be a waste of pages.  Instead I will endeavor to equip the reader with the knowledge that will allow you to explore these elements of reality for yourself. 
One must believe, or at least be ready to believe.  As an overabundance of skepticism will prevent the mage from leaving the body and entering the alternate reality.  One must achieve an altered state of consciousness. The best method of course is meditation or controlled breathing.  By doing so the mage, through visualization, leaves the body and enters the astral.  There are many techniques for this most difficult of transitions, the most simple of which is meditation and visualization of stepping out of one’s body and being reborn into an alternate world.
This skill is much more difficult than most other magicks due to its subjective nature.  Many times the beginner will question if actions taken in the astral really happened, or if they were a hallucination or mental fantasy.  Sometimes they will be real, other times they will be the products of an imaginative mind rushing towards a delusion of success.  It is important to remember that lucid dreaming is also a viable method of gaining access to the astral. 
The best resources on astral magic and travel are the following two books:
Arcane Lore – A text that focuses on gaining access to the astral via lucid dreaming.
The Magical Use of Thoughtforms: An experiential and academic combination of the techniques and descriptions of the above two books, very easily the authority on the subject.