A long time ago, on Tatooine...
...Computers in those days were based on archaic vacuum tube technology. The basic principle involved the use of a charged field to control and amplify the flow of electrons through a vacuum, and the technology was kind of at a standstill.
Anyway, an enterprising band of jawas was sick and tired of dealing with complicated, power-guzzling, ENIAC-sized behemoths taking up three-quarters of their sandcrawlers. After some hard work and long hours in the laboratory, some of their clever engineers perfected a method to control the flow of electrons in a solid silicon crystal, instead of in a vacuum -- yielding a device they dubbed "utinni" but which we know today as a "transistor".
The transistor was small, didn't need to heat up, and operated on a fraction of the power of the jawas' existing hardware. Within a few years of the breakthrough the jawas created palm-sized transistors, and sixteen months after that they produced the galaxy's first true integrated circuit, which combined several million tiny transistors onto a single microscopic silicon slab.
The first computer system run by this hardware was rather mundane. Its purpose was to keep track of every droid transactions for each sandcrawler in the fleet. The system was successful because it was relatively easy to use, and because it freed jawa accountants from the burden of spending countless hourse sorting through reams of incomplete, printed documentation. The program was known as the Automated Robot Tracking and Organizing Operation (ARTOO for short), and it was soon deployed on every crawler in the fleet.
The 3.0 version of ARTOO came with an important new feature. A handy "comlink" function was included with the upgrade, allowing computers on board different sandcrawlers to share data and communicate with one another virtually seamlessly.
Never before had the jawas been able to maintain their finances so efficiently and meticulously. Now they had an on-line record of every purchase and sale they made on Tatooine. How wonderful, for example, it was for them to be able to produce, at the touch of a button, a precise, detailed record of the sale of a couple scavenged robots to, say, a local farmer and his nephew.
Although it meant that a few jawa paper-pushers soon were out on the sand, the system was hailed as a great success and paid for itself several times over within the first year of its release.
Naturally, the Empire was bound to find out about the jawas' advances sooner or later. Tatooine legend has it that the "look sir, droids" trooper was the first to sneak into a sandcrawler and see the technology first-hand, but this may be fabrication. In any event, news of the invention reached the ears of Private First Class Tarkin, who, upon recognizing the golden opportunity, sent a legion of troops to Tatooine to steal the technology.
Once the hardware was in his hands, Tarkin engineered a slick program in which various Imperial corporate entities and commercial vendors were "seeded" with the jawa hardware and software.
Second Lieutenant Piett was tasked with creating a falsified "history" of development for the Imperial hardware engineers, in order to obscure the true source of the breakthrough. (The fruits of these projects would eventually lead, directly or indirectly, to cloaking technology, bulletproof stormtrooper armor, fiberoptics, and of course the Empire's infamous Strategic Defense Initiative.)
The fundamentals behind the jawas' droid software were also hijacked by the Imperial scientists. Tarkin liked the way computers in different sandcrawlers could communicate with each other. He envisioned a similar network for the computers on-board the proposed "Death Star". For instance, he reasoned, how efficient it would be for the seven computers controlling the station's tractor beam to be linked together by one master system.
A few years later, although Tarkin didn't live to see it, Piett took the project to its logical conclusion, directing his engineers to create a "network of networks" which would allow all the Star Destroyers in the Imperial Fleet to "speak" to one another. Sith Lord Darth Vader, who hadn't tinkered with computers since he was a little boy, later requested a graphical "front-end" to assist him in using the software. And thus the World Wide Web was born.
Disclaimer: This story is based on characters and situations created and owned by Lucasfilm, Ltd. No money is being made and no copyright infringement is intended.