Brilliant Pebbles = killer robot rocks
In layman's terms, Brilliant Pebbles, as the name suggests, were basically supposed to be man-made rocks that would be seeded into low orbit and packed full of micro-electronics, high-tech sensors, and thrusters. These rocks would lie in wait in orbit, conserving energy, until their sensors detected the launch of a Soviet ICBM. Then they would supposedly spring into action automatically and autonomously, activating their thrusters to maneuver themselves into the path of the enemy missiles during their "bus" phase in order to destroy them or at least throw them off course or prevent them from dispersing their multiple nuclear warheads.
The Brilliant Pebbles concept had initially been conceived of by Lowell Wood of the Livermore Lab in 1986, and was aggressively marketed to the Reagan administration by Wood and his mentor (and extreme SDI enthusiast) Edward Teller. It took several years for the idea to gain traction, but by 1989, with George H.W. Bush set to assume the presidency, the Brilliant Pebbles were seen as a way to save SDI from a much more skeptical new administration, and thus were strongly endorsed by Strategic Defense Initiative Office (SDIO) director James A. Abrahamson in his farewell report. By this point, SDI was on the verge of failure, as it had become clear that earlier "Star Wars" concepts like mounting X-Ray lasers or rail guns on orbital battle stations were at best many decades away from feasibility, and at worst, an absurd fever dream.
The main perceived advantages of the Brilliant Pebbles were that that they would be vastly cheaper to produce than thousands of heavily armed orbital battle stations, and also that they might actually be producible in the near future using existing technologies. Accordingly, the Brilliant Pebbles were seized upon by those desperate to save the SDI program and its absurdly high levels of research funding, even though Pentagon officials admitted internally that it was the equivalent of a "Hail Mary" pass in the dying moments of a football game.
In fact, the Brilliant Pebbles concept did succeed in extending the life span of the SDI program throughout the four years of the Bush administration, burning through many more billions of taxpayer dollars. But unfortunately it turned out that existing technology was not quite as awesome as Teller and Wood had asserted, and when the Bush administration demanded a realistic version to be ready by the scheduled deployment date of 1993, the original idea of five pound "pebbles" propounded by Wood and Teller rapidly expanded into 100-pound, 3-foot-long boulders, which instead of costing $100,000 each, were now estimated to cost as much as $1.5 million apiece, for a program that was calling for 4,600 "pebbles" to be in space at all times, and that was before factoring in launching costs!
Nevertheless, even the new, more realistic, much bulkier pebbles still promised to cost much less than any previous SDI proposal, so the Bush administration kept pouring money into them and promising a deployment date in 1993. But in the meantime, the Cold War came to an end, and the fabricated "success" of the patriot missile against Iraqi SCUDs in the Gulf War turned attention away from space missile defenses back to land-based missile interceptors. Meanwhile, the Pentagon had grown increasingly leery of loading up Earth's orbit with thousands of autonomous killer robot rocks. Thus when Bill Clinton assumed the presidency in 1993, he quietly killed the Brilliant Pebbles program, and renamed the entire SDIO to the "Ballistic Missile Defense Organization" (BMDO), signalling the new emphasis on land-based anti-missile defenses.