It is a mysterious satellite, of unknown origin, discovered in 1960 which shadowed Sputnik. It is believed to have been of extraterrestrial origin, and signaled back old radio waves from the 1920s and 1930s before it disappeared. In short wave patterns analyzed by astronomer Duncan Lunan, it revealed its origin as Epsilon Boötes (or the star system as it was 13,000 years ago).
A report I just read about it from the news dated March 7th 1960:
"Three weeks ago, headlines announced that the U.S. had detected a mysterious "dark" satellite wheeling overhead on a regular orbit. There was nervous speculation that it might be a surveillance satellite launched by the Russians, and it brought the uneasy sensation that the U.S. did not know what was going on over its own head. But last week the Department of Defense proudly announced that the satellite had been identified. It was a space derelict, the remains of an Air Force Discoverer satellite that had gone astray."
The only reason it was worth note was that at the time we couldn't accurately track even large space debris - if a satellite wasn't broadcasting, it could only be tracked when it flew over certain facilities, which wouldn't happen every orbit, leading to a lot of disconnected readings. It was picked up by a brand new satellite tracking system, and was the first unaccounted for object the system detected, the second being about two days later (being another satellite in the same series).
By 1960 there were already four organizations using three facilities to launch satellites, and only one of those organizations was in the general practice of making its launches known. A high number of satellites failed during launch, leaving no clear indication if they ended up in orbit or not, and many were little more than radio beacons or radar reflectors to begin with.
Now, a few things about the conspiracy theory: The satellite didn't shadow sputnik. It wasn't even in a similar orbital altitude or plane - it was in a near-polar orbit. The conspiracy theory says neither the US nor USSR had a successful polar orbit launch by that time, but in fact NASA had made 10 near-polar launches, the USSR had t least two, and the US Air Force admitted to 5 after this particular blunder.
Now, right after it was found, HAM radio operators claimed to start picking up strange signals. The signals were consistent with the sort of electronic noise you can generate with an oscilloscope, which is a prank that's made fools of HAM operators as recently as August this year. One person, with no cryptology experience or education, claimed to decode the message. With no astronomical experience or education, he claims he immediately recognized it as a series of coordinates that produced a star chart which matched what the sky would look like from a particular star several thousand years ago (despite the fact that astronomers couldn't make accurate enough distance measurements to identify such a chart).
Having made such an amazing discovery, he immediately destroyed the hundreds of pages of math he needed to decode the message as well as the decoded star chart and only announced his findings over HAM radio on the same frequency the signal had been detected on. When later provided with the original coded message, he couldn't remember how to decode it.