The frequency of a simple light wave is directly related to its colour (cf. spectrum). For the lowest frequencies of visible light, that colour is red, light of the highest frequencies appears blue. If the frequency of a light wave is shifted towards lower frequencies (for instance by the doppler shift), that corresponds to a colour shift towards the red end of the spectrum, and is hence called a redshift. Consequently, a shift towars higher frequences is called blueshift.
From this, “redshift” has come to acquire a more general meaning. It is used to denote any shift towards lower frequencies, even for types of electromagnetic radiation where the frequencies do not correspond to any visible colour, and more generally still, for other types of waves as well (for instance for gravitational waves).
In the context of general relativity, the gravitational and the cosmological redshift are of particular interest.