As the name of a color, violet (named after the flower violet) is used in two senses: first, referring to the color of light at the short-wavelength end of the visible spectrum, approximately 380–420 nm when indigo is recognized as a distinct spectrum color, or more commonly 380–450 nm (this is a spectral color). Second, violet may refer to a shade of purple, that is, a mixture of red and blue light, and not a spectral color (see a discussion of the distinction between violet and purple). Spectral violet is outside the gamut of typical RGB color spaces, and although it can be approximated by that color shown below as electric violet, it cannot be reproduced exactly on a computer screen.
For the record, this means nothing to me.
Indigo is the color on the electromagnetic spectrum between about 420 and 450 nm in wavelength, placing it between blue and violet. Although traditionally considered one of seven divisions of the optical spectrum, modern color scientists do not usually recognize indigo as a separate division and generally classify wavelengths shorter than about 450 nm as violet. Those who do accept indigo as a separate spectrum color regard its wavelength as being from approximately 420 nm to 450 nm.
Like violet, whether indigo is considered a shade of purple depends on context. Common English usage defines a purple color as any color between red and blue whereas in color theory, a purple color is defined as any non-spectral color between violet and red (not including violet or red since they are spectral colors). Thus indigo and violet fit the common but not the color theory definition of purple.
One can see spectral indigo by looking at the reflection of a fluorescent tube on the underside of a non-recorded compact disc. This occurs because the CD functions as a diffraction grating, and a fluorescent lamp generally has a peak at 435.833 nm (from mercury), as is visible on the fluorescent lamp spectrum.
The first recorded use of indigo as a color name in English was in 1289.
So that sorted that out.