There are two terms for this technique: overdubbing and multi-track recording. These techniques are not new, and they are used on practically all music recordings everywhere all the time.
These techniques are not only used for vocals; they are used for all musical instruments, and for sound effects in video and films as well.
It is safe to say that virtually all music recordings and movie soundtracks that you have ever heard have made use of overdubbing and multi-track recording.
Overdubbing has been in use since the 1920s, when recordings were made on lacquer discs, and later on reel-to-reel tape recording machines, which were developed in the 1940s. The technique of overdubbing was accomplished with two recording devices and a sound mixer. The musician most associated with pioneering the use of overdubbing in the studio, starting in the 1930s, is the American guitarist Les Paul (1915-2009). His landmark album The New Sound, released in 1950, featured him as the sole musician playing all the instruments, in multiple musical parts, recorded one at a time and overdubbed onto each other. Later Les Paul produced many hit singles featuring his singing partner Mary Ford, overdubbing harmony vocal parts to make the sound of vocal duets, trios, quartets and choruses.
Multi-track tape recording machines were developed in the late 1940s, and became commercially available in the mid-1950s. Again, Les Paul is the person most associated with pioneering their use. Once multi-track tape recording machines became common in recording studios, the practice of multi-tracking and overdubbing became the standard method of making all types of music recordings and motion picture soundtracks, to this day.
Nowadays multi-track recording is accomplished on computer-based systems which no longer utilize recording tape. Almost any computer program that can record audio (a digital audio workstation, or DAW) can accomplish multi-tracking and overdubbing.
Multi-track recording on Wikipedia
Overdubbing on Wikipedia
Les Paul on Wikipedia
As other answerers have pointed out, the YouTube example you provided gives an example of a special use of overdubbing called double-tracking. Double-tracking is the technique whereby one singer records his or her vocal part twice, attempting to sing exactly the same thing both times, in order to produce a "thicker" vocal sound. This technique has been around for some time; I believe it became popular when it was used by the Beatles on their early recordings in the early 1960s, although the Beatles did not invent the idea.
Double-tracking is also commonly used with string orchestras, so that, for instance, eight string players on a recording session can be made to sound like sixteen string players (although this has always been controversial since it means that the producer can achieve the effect by paying eight musicians to play instead of paying sixteen musicians).
Double-tracking can be simulated by having one single recording electronically processed to make it sound as if there are two very similar recordings, with the use of electronic modulation through the techniques known as phase-shifting, chorusing, and delay.