To most of the ancient cultures of the Mediterranean and Middle East, where so much of Western cultural heritage comes from, there was no such thing as coincidence. Events in the visible world were the result of decisions and events in the invisible, supernatural world. This way of thinking combines two fundamental malfunctions of human cognition: our tendency to see agency and causation everywhere. In such cultures, if you wanted to understand what was going on in the world, and most crucially, gain any kind of insight into the future, you needed knowledge of the supernatural world. This knowledge was obtained through divination.
Most of these cultures recognized two different kinds of divination: technical and non-technical. Non-technical divination was the "divine madness" described in Plato's Phaedrus: prophecy, dreams and visions. To the Hebrews, prophecy became the only legitimate kind of divination when the casting of lots started to be frowned upon, and some of this prohibition survives to our time in Christianity. Especially in the charismatic forms of the faith, God may well send visions and prophetic dreams to believers, but ouija boards and tarot cards are considered demonic.
Techical divination, on the other hand, relied on inductive reasoning. Patterns would be found in the stars, in the fall of lots, or in the entrails of sacrificial animals, and the will of the gods would be deduced from these. To our sensibilities, this seems crazy, but in a way, technical divination is one of the forerunners of science. Some cultures, especially in the ancient Middle East, developed corps of technical diviners and elaborate institutions of divination, like astrology and haruspicy, with standardized practices and extensive documentation.
In the clearly magical universes of most fantasy stories and games, divination tends to be conspicuously absent. Gods are either considered withdrawn from the world altogether, or even when they aren't, their purposes and plans are so inscrutable that trying to understand them is futile. At best, most characters seem to be deists: even if they believe a god or gods have created their universe, it still essentially functions according to secular, scientific logic. In order to intervene, even gods will often need to physically manifest themselves and become directly involved. In general, fantasy worlds follow a very modern logic.
In my personal opinion, this is because none of us outside closed mental institutions actually believe in the supernatural any more. There's a wonderful Geoffrey Chaucer blog on the net with a medieval version of Snakes on a Plane, called Serpentes on a Shippe. It's a comic retelling of the movie - up to a point. What follows is technically a spoiler, so if you haven't done so already, do read the story.
In the movie, the protagonists use their various abilities to defeat the snakes. In the medieval version, they realize that mere humans can't triumph against evil on their own, and pray to God and Saint Patrick, who smite the snakes and rescue the protagonists. This is an authentic medieval ending: the point of the story is its sensus spiritualis, which teaches the reader that real power is only wielded by God. Imagine a Hollywood movie with that ending! Or, for that matter, try running a tabletop role-playing campaign where the player characters can't defeat their enemies by force or solve mysteries and puzzles by logic, but must prostrate themselves before a god and beg for divine intervention, and give thanks when they get it. Most of us would consider it a terrible campaign.
Some of this is an effect of our storytelling conventions; we like stories about exceptional individuals who triumph over adversity thanks to their personal qualities or skills. But so did our ancient and medieval forebears. The key difference is that unlike them, we don't believe the world is governed by the supernatural. (If they even did; we really know almost nothing about what ordinary people thought in the distant past.) The protagonist or player character needs to make their way through the woods by their skill in orienteering, not through divine guidance. If the climactic shootout in a western ends in God smiting the bad guy, we feel cheated of the proper resolution of the story, and it just rings profoundly false to us. We know that in real life, gods don't smite anyone. Some of us say they believe otherwise, but if you look at the way they live their lives, they follow a completely secular, worldly logic in all practical things. Our problem-solving completely rejects the supernatural.
In terms of the dual processing theory of reasoning, in formal problem-solving situations like a puzzle or mystery in a tabletop role-playing game, we use our analytical reasoning, which is highly resistant to the errors of the primary system that make us assume supernatural agency and causation. So we don't come up with supernatural solutions to practical problems.
This creates a unique problem when we try to role-play magical thinking. Our analytical thinking is so firmly secular that we don't even realize it. I don't know if other GMs have had dramatically different experiences, but very few of my players have ever attempted to make any real use of divination, even when explicitly provided for in the setting and system. My hypothesis is that this happens for two reasons: both our anachronistic analytical thinking, and the fact that getting an answer or solution from an outside source feels like cheating. I think that to some extent, players using divination will feel that they haven't put in the work to get the answer, so being given it directly is unsatisfying. Providing non-inductive divinatory experiences for players can, at worst, feel like a ham-handed attempt to railroad the campaign by telling them what the GM wants them to do.
In my Rogue Trader campaign, I wanted to take the fanatical religiosity of the Imperium seriously, and one of the aspects I wanted to explore was divination. This would incorporate non-technical divination like dreams and omens, but I also wanted to experiment with providing players a means of technical divination. I wanted to try two strategies that would make technical divination more interesting and satisfying to players. Firstly, have something concrete for them to use. This makes the divination into a bit more of an occasion, and hey, who doesn't like props? Secondly, tie the divination to player character skills, and leave the interpreting to the players! This should counter the notion that divination is getting something for nothing, and instead of appearing as a case of divine intervention solving problems, it should give players new information to add to their reasoning.
A rare example of a game where I thought technical divination was used intelligently was the wonderful Quest for Glory IV, which featured several Tarot readings and left me with a permanent itch to incorporate the Tarot in a tabletop role-playing game. Therefore, my campaign includes the Emperor's Tarot.
Luckily, the Emperor's Tarot hasn't been exhaustively described by Games Workshop anywhere, so I took considerable liberties in designing the cards. I omitted pretty much all the cards referred to in various GW works, if only because I find Games Workshop's bizarre pseudo-Latin absolutely intolerable. There are four suits of minor arcana, each named for a Segmentum: the Pacificus, Tempestus, Obscurus and Ultima. The major arcana are the Solar cards. Each suit has a corresponding element and color. My partner created the physical cards, and I think they're absolutely brilliant: they're big, solid and impressive-looking. To illustrate the first version, we chose images from a Milo Manara tarot deck we own, as well as art from Luis Royo and some miscellaneous images. There is a notion of eventually producing a physical deck of directly 40k-themed cards, but since I can't draw, this depends on the industriousness of the artist.
Below is a picture of the whole deck, as well as its first ever use in my campaign: a Tarot reading done for a character visiting a fortune-teller's shop on other business. Fans of Quest for Glory IV will recognize the spread.
For readings done by NPCs, I had the NPC explain each card and provide an interpretation. This was fun enough and made for some good scenes, but obviously my real goal was to get my players to use the cards themselves. In Rogue Trader, each Navigator starts with a Tarot deck, so introducing them was easy. What I did was create a small booklet explaining how to do a spread, and what meanings each particular card has. This is quite easy to do; the laziest way is to look up the standard interpretations of modern-day Tarot cards from Wikipedia! Each player competent in divining was given a copy, and told that once a session, they could use their Trade: Soothsayer or Psyniscience skill to procure a reading on a topic they requested. For each session, I would prepare a few spreads on topics I was prepared to offer information on. Remember that the cards themselves should indicate what the subject is, and this should also be an interpretative challenge! Then when a player wants to use the Tarot, you stack the deck to produce the reading you prepared and hand it to them. (bonus points for doing a fake shuffle first!)
Below is an image of the first player-initiated use of the Tarot in my campaign. My main group of players, led by the Rogue Trader himself, was investigating some strange goings-on at a penal colony. Everything seemed to be in order, but our Navigator wasn't buying it, and asked for a reading. I gave him a pre-stacked deck, and he got out his interpretation guide and laid out the spread.
To really make using the Tarot interesting, don't create readings that have obvious meanings. Your deck shouldn't have a card for Ork that literally only means ork, and always signifies that there are actual orks involved in what's going on. This way, your players will have to actually try to interpret the reading in light of their other information and expectations, and this can produce results you never anticipated. If players start fixating on certain interpretations of cards, this becomes a wonderful way to misdirect them.
In this particular case, one of the main cards that came up was the Demon, accompanied by the Magus and the Tomb, the last of which denotes, among other things, buried secrets. These thoroughly convinced our Navigator that there was something sinister going on, and he absolutely insisted that they must investigate further. This ended up having a strong effect on the session, eventually leading to a smuggling ring being exposed and several colony officials being shot. This reading was an answer to a fairly straightforward question, and produced a very straightforward reading, but I felt it was an excellent and very thematically appropriate addition to the session.
Overall, I've found incorporating the Emperor's Tarot into my campaign to be very worthwhile: not only were the cards fun to design and make, but on several occasions, I've felt they've added a whole new dimension to my players' problem-solving and, hopefully, their enjoyment as well. I would encourage anyone running a game in any kind of magical universe to incorporate some sort of divination in their campaign.
My notions of divination here are based on Professor Martti Nissinen's lectures on Old Testament exegetics at the University of Helsinki. A pretty good introduction is Nissinen, Martti: Prophecy and omen divination: two sides of the same coin, in Annus, Amar (ed.): Divination and Interpretation of Signs in the Ancient World, The Oriental Institute of the University of Chicago, Oriental Institute Seminars n:o 6, Chicago, IL, 2010. Any distortions or errors are obviously my responsibility, and the application of divination to role-playing, as well as the juxtaposition of modern-day secular logic with the magical thinking of divination are my personal thoughts.