I've never quite understood this about the series. They have space travel, but most other things about their universe seem primitive compared to ours.
It is connected to the Buthlerian Jihad - the crusade against computers and AI technologies that left them universally banned. The punishment for violation is death.
From the glossary in the original Dune:
Jihad, Butlerian: (see also Great Revolt) — the crusade against computers, thinking machines, and conscious robots begun in 201 B.G. and concluded in 108 B.G. Its chief commandment remains in the O.C. Bible as "Thou shalt not make a machine in the likeness of a human mind."
They are using mentats, the human equivalent of a computer, which came with analytical capabilities - the equivalent of programming, and with mnemonic abilities - the equivalent of computer data storage (databases).
For space travel they are using navigators, which are also an equivalent for a specialized navigation and prediction computer.
One of the prequel books by Herbert's son: Machine Crusade, fills in the past where computers took over the early planets settled by humanity. The uprising to overthrow the computer/cybernetic/robot civilization was called the Butlerian Jihad (one of the final battles involved glassing and sterilizing Earth). It was so traumatic that following the destruction of the computers, having computers, or even machines that worked like human minds, were forbidden.
The series of novels isn't entirely consistent about what exactly constitutes a computer and what sort of limitations are imposed by the Butlerian Jihad prohibitions.
In Dune, it first seems that the limit stops at a what we would recognize as artificial intelligence able to pass a Turing Test - in practical terms where a human interrogator unable to see subjects cannot tell the differences of written (or typed) responses to questions asked of both a human and computer. This would allow for data storage, decision trees (if/else statements), and automated control systems using state machines. One would think that this would be necessary for signal processing required for interstellar communication at the very least. Basically, at Dune, it does not appear that Turing Completeness is punishable by death :-)
Herbert would appear to raise the bar in God Emperor of Dune. Moneo (Leto II's lieutenant) is observed by Duncan Idaho to be hiding printouts from something behind a hidden door of what Duncan assumes to be a computer used for simply storing data, prompting Duncan to speculate that Moneo is 'testing the boundaries' of the Butlerian Jihad prohibitions. This would indicate that any sort of digital storage device constitutes a computer "in man's likeness"...now HTML (in our day) would be forbidden. This is fine so far because it makes accomplishments of biological and mechanical engineers even more impressive, and the scope of technologies used in the Dune universe is limited at this point to what is seen on Arrakis itself, which is fairly primitive but fascinating.
However, in the next book Frank Herbert takes a detour in Heretics where he goes off-planet and describes devices like the T-Probe, torture/interrogation devices which can basically hold a subjects consciousness in digital memory, and interrogators can query and manipulate the digital representation of consciousness and make modifications to the subject. These never raised any questions or seemed alarming in concept, however at this point in Heretics people had more important things to worry about...
So, it jumps around a bit but I would speculate that, much like our own religions, the rules are interpreted differently by folks over time.
Spoilers ahead, hover to see them:
Because machines that think were extinguished during Butlerian Jihad because they represent a perverted imitation of human mind. They use mentats (human computers) instead. However, if I'm not mistaken, Ixians break this rule, and other worlds tolerate this mostly because they need their technology.
It's one of the most interesting things about the series. As others have noted in their sound answers, the Butlerian Jihad, fought against machines who tried to take over human society, resulted in a universal ban of what we might call AI.
Despite this, however, the planet Ix, and to a lesser extent Richese, are seen as leaders in technology and are somewhat mistrusted because of their flirtation with the boundaries of the anti-AI rule. Indeed, in the later books Leto sees the rebirth of AI on Ix as the possible downfall of mankind, but doesn't fully defeat them. Dancan Idaho (reincarnated) thinks it's because Leto can't help but be fascinated by the possibility of a joint man-machine future.
Idaho: "He was fascinated by the idea of human and machine inextricably bound to each other, each testing the limits of the other."
This exploration of the idea is about as far as the AI aspect of Dune gets though.
It's one of the core assumptions of the series that there is no AI, but it does come up semi-frequently. By restraining this particular aspect of technological advancement, the idea of a feudal society with Dukes and Barons, hand-to-hand war fighting, concubines, etc. becomes credible.