The most compact description of Alexandrian Tradition can be found by considering the two answers to the question, How many Alexandrians does it take to change a light bulb?
One answer is, How did the Gardnerians do it? Which illustrates that the bulk of material passed down in the Alexandrian Tradition is pretty much identical to the Gardnerian material, barring a few transcriptional errors. This is not surprising, considering that we now know that Alex Sanders, the founder of the Alexandrian Tradition, was initiated into the Gardnerian Tradition by Pat Kopanski, an initiate of Patricia and Arnold Crowther. Although Alex claimed to have been initiated into Wicca by his grandmother, what he taught was essentially Gardnerian Wicca, and no one really believes the grandmother story anymore.
The other answer to the light bulb question is Thirteen: the High Priestess, the High Priest and the first eleven off the street. This answer illustrates a key difference in attitude between Gardnerian and Alexandrian Tradition: where Gardnerian Tradition stresses secrecy and caution (perhaps wisely so), Alexandrian Tradition tends to be more open and accessible, sometimes to the point of recklessness. Alex Sanders was notorious for meeting people in a pub, chatting with them over a pint, and initiating them that same evening. It is perhaps no accident that the most complete published version of the Gardnerian/Alexandrian Book of Shadows was published by two of Alex's initiates, Janet and Stewart Farrar. Alex's attitude seems to have been "Initiate them all - let the gods sort them out."
Of course, modern Alexandrians don't necessarily go about things the way Old Alex did. Some Alexandrian covens feel it is important to preserve the tradition exactly as it was handed down to them, with neither omission nor invention. These covens tend to focus on the written material in the Book of Shadows and to emphasize the similarities between Alexandrian and Gardnerian Traditions, while downplaying the unique ethos Alex brought to his practice of Wicca.
Other Alexandrian covens preserve the original material, but place more emphasis on continuing to develop new material, so that the tradition might grow and change over time. These covens focus more on the dynamic energetic current of the tradition, and tend to regard the Book of Shadows as a work in progress, or a record of the ongoing growth of the tradition.
In general, Alexandrian Tradition tends to emphasize ritual form, and each coven tends to stick to an established ritual framework rather than coming up with a completely new ritual each time. Within that framework, there may be specific points where improvisation based on the inspiration of the moment is allowed; but the ritual structure at least makes it clear whose turn it is to improvise. As a result, Alexandrian training tends to focus first and foremost on ritual skills: learning to follow the form and fit into the energy pattern of a ritual. Some covens also teach practical magic, such as divination, spellcraft, herbal magic, etc.; some covens teach their own particular flavors of thealogy: the nature of the Gods and other spiritual beings, beliefs about reincarnation; but these vary considerably from coven to coven.
History and contact information for a variety of traditions of English Witchcraft can be found at The Beaufort House Trad List.
Previously published material that would typically appear in an Alexandrian Book of Shadows can be found at the Public Domain Alexandrian Book of Shadows.