The name that set off the Space Race was not a code name or a boast. It was the everyday Russian word for a satellite, and at heart it means a companion on the road.
The Soviet Union launched Sputnik 1 on October 4, 1957, from the site now known as Baikonur. It was a polished metal sphere 58 centimeters across, weighing 83.6 kilograms, that broadcast radio beeps. The batteries ran out after 22 days. The satellite burned up on January 4, 1958, after about 1,440 orbits.
Etymonline breaks the word into three parts. The prefix s- means "with" or "together." Put' means "path" or "way," going back to a Proto-Indo-European root meaning "to tread, go." The suffix -nik marks a person or thing tied to something. Put together, a sputnik is someone who shares your path: a fellow traveler. The full phrase was sputnik zemlyi, "fellow traveler of the Earth."
That is exactly how the English word satellite started too. Latin satelles meant an attendant or bodyguard, someone who stays close to a greater person. Johannes Kepler applied it to the moons of Jupiter in the 1610s. Russian and English reached for the same image, a faithful companion, and simply used different languages to express it.
In English, sputnik quickly became a general term for an artificial satellite. It also had a surprising side effect on the language. English had already picked up the suffix -nik from Yiddish words such as nudnik, but Sputnik made it wildly productive. When American satellite attempts failed, newspapers called them "flopnik," "kaputnik" and "stayputnik."
The most lasting coinage came in 1958. San Francisco columnist Herb Caen joined the Beat generation to the fashionable suffix and produced beatnik. A word for a Soviet satellite ended up naming American poets in black turtlenecks.
Not everyone in Washington was shocked. Wikipedia notes that President Eisenhower had been forewarned of Soviet rocket capabilities by U-2 spy plane photos, and he responded calmly. What he misjudged was the public reaction.