What is the definition of a noiselet?
The wavelet transform has the desired property of presenting most images as a sparse signal where most of the signal energy is concentrated in few coefficients.

A noiselet is structured to take such sparse representation and generate a noise-like signal from it. The coefficients resulting from the noiselet transform are spread out to almost equal magnitudes across the whole signal.

Why are they useful?

noiselets are are highly 'incoherent' with the signal representation in the wavelets domain. This means that noiselets can be used as a sensing matrix to recover sparse signals with high probability of success. Section 2.2 in this paper provides some mathematical details and an image recovery application.