The contribution of the Hermetic Order of the Golden Dawn to our modern understanding and use of the tarot cannot be overestimated. Here is a link to a brief article on Biddy Tarot by Richard L. Palmer that lays out the fundamentals.
The esoteric concepts and structures devised by the Golden Dawn and its principal architect, Samual Liddell “MacGregor” Mathers, underlie the two most famous decks in use today: the Rider-Waite-Smith and the Thoth. In my own experience, the chief value to be found in the Golden Dawn canon resides in the system of elemental, astrological, numerological and qabalistic correspondences developed by Mathers and his cohorts, and in the method of Elemental Dignities for assessing the potency and ease of expression of a card according to its interaction with adjacent cards, both of which are described in “Liber T” of the Golden Dawn Knowledge Papers. A reconstruction of Liber T with a broader outlook was developed by James Eshelman and published as “Liber Theta,” available free from the College of Thelema website. Robert Wang’s book, The Qabalistic Tarot, is another valuable resource.
Here are two charts created by “Richard” of the Aeclectic Tarot forum (who may or may not be the Richard L. Palmer mentioned above) that lay out the arrangement of correspondences between the Thoth Tarot and the Chaldean system of astrological decans. Note that only the twelve zodiacal trump cards are assigned places on the innermost circle of the second chart; the seven planetary and three elemental trumps are not represented. (The zodiacal and elemental attributions for the planetary trumps are derived from the traditional sign rulerships of the assigned planets.) In addition, the Aces and the Princesses/Pages are aligned according to polar quarters and not individual signs, as shown in the second chart; each set relates to three signs and is centered on the fixed sign of the matching element in its respective quarter. A composite wheel encompassing both charts – but without the Thoth titles and trump card associations – can be found in Robert Wang’s book.