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Photon noise is variation in the measurement of electromagnetic radiation due to its quantum nature, i.e., if the number of incoming photons is small, the rate at which they trigger a detector can vary through pure chance. The term shot noise is also used for such photon noise as well as the somewhat-analogous tiny variation in electric current due to its quanta, i.e., individual electrons. The concept of such noise was introduced in 1918 by Walter Schottky.
In astronomy, clear, unambiguous observation of distant objects may be limited by photon noise. The noise in a finite observation follows a Poisson distribution (thus it is also known as Poisson noise), which with enough photons, is virtually Gaussian (i.e., the normal distribution described by the Gaussian function). However, astrophysics looks to make use of whatever small amount of data you can acquire, so disentangling the photon noise (and thermal noise) from the signal is of interest, and efforts are made to model the various components mathematically to untangle signal from noise. In general, more observation time increases your ability to isolate the signal from the photon noise. The term shot noise level (or Poisson level) refers to the degree of photon noise that is contributing to the resulting signal/noise mix.