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Wave number counts cycles per unit length, so it is the reciprocal of wavelength. Physicists usually mean the angular form 2pi/lambda, while spectroscopists mean 1/lambda and quote it in reciprocal centimetres. A 500 nm wave is 2.0e6 per metre, or 20,000 cm-1, and 1.26e7 rad/m in angular terms.
Angular wave number
k = 2 pi / lambda
Spectroscopic wave number
nu-bar = 1 / lambda, usually in cm-1
Because it is proportional to photon energy and gives convenient numbers - the infrared fingerprint region runs 400-4000 cm-1.
Both, in their own field. The factor of 2pi between them is why the units are quoted as rad/m for the angular form.