Any tape system (except closed-loop cartridges) requires two reels: one (the supply reel) to hold the tape before it is played, and another (the take-up reel) to receive tape spooling off the first reel. In most systems, each end of the tape is permanently attached to a reel and the whole system ensconced in a portable case. ReelToReelTape, or more precisely, open-reel, system, you carry around only the supply reel. When you're ready to play you have to attach the free end to a take-up reel that (hopefully) is already mounted on the tape machine. The 1/2-inch mag tape associated with 1960's computing worked like this, although some machines had a built-in take up reel and were sophisticated enough to suck in the leader on the tape you inserted and thread itself.

When I worked as a radio announcer we used open-reel tape regularly. Some tapes were marked WTO (wound tail-out) meaning that the tape had been played to the end and had to be rewound before it could be played again.

Domestic stereo reel-to-reel tape decks were usually 1/4-inch tape using four tracks. The top and third tracks carried the left/right channels; the tape could be turned over to use the other tracks. This track interspersing increased crosstalk between the two recordings, but made the design of the tape heads more practical. Tape speed for "hi fi" recordings was often 7 1/2 inches per second, with decks often offering additional speeds of 15 ips, 3 3/4 ips and sometimes 1 7/8 ips (which was later used as the tape speed for cassettes).

In the late 60's record companies were releasing albums on 33 1/3 rpm vinyl, 8-track cartridge, and reel-to-reel 1/4 inch tape at 7 1/2 or 3 3/4 ips - although not all vinyl releases made it to reel-to-reel.

ReelToReelTape was used in recording studios up until very recently, when digital recording finally caught on.  It has also been used in the composition of Musique Concrete, an abstract and experimental style of music that performed transformations on the physical tape, (such as stretching, cutting and flipping, cutting and reversing, etc.).  ReelToReelTape is to the Ambient/Experimental Musician, as Vinyl to the Hiphop DJ.

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''I remember albums on reel-to-reel--a friend of mine's parents had them, when I was a kid in the '70's.  Vinyl has the obvious advantage of random access (and hip-hop DJ's began to use this feature in the early '70's), and eight-track tapes were portable (in cars, for example)--what are the advantages of reel-to-reel?''

It was much better to handle and considerably better in fidelity than the wire recorders which preceded them. (Have you ever tangled string on a rod and reel when fishing?). I remember the Webcor Wire Recorders which were in use in the 40's. ReelToReelTape recorders were the natural evolution from the metalic media to the plastic media. But they were used as digital recorder/players today to convert between representations of sound to the objective sound via transducers. 

Audio quality was the single advantage.

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