Gene expression is inherently noisy. This means there is variability in the gene's response to an input. Say a population of genetically identical cells all receive the same input signal (e.g. concentration of an inducer like IPTG, light, or a stressor like heat). If you measure the response of each cell to the input, their responses will vary across the population. Some will respond too much, some too little. This variability is called noise. Some of the noise is Poissonian and is a result of the discrete nature of reactions in cells.
Cells typically only have a single copy of a gene. So, when an input signal arrives, the gene will start making mRNA which will be translated into protein. Because there is only one gene, mRNA must be made one at a time (discretely). The Poissonian description of this noise means these independent, uncorrelated, discrete events are spaced in time according to an exponential distribution.
This variation was explored by Michael Elowitz et al. in the 2002 paper Stochastic Gene Expression in a Single Cell. This important paper was one of the first to study noise in gene expression. Elowitz et al. wanted to understand how much variation in gene expression was intrinsic (specific to a particular gene) and how much was extrinsic (variations in global assets like polymerase and ribosomes). So, the authors crated a plasmid (injectable sequence of DNA) with cyan fluorescent protein (CFP, on the green channel) on one side and yellow fluorescent protein (YFP, on the red channel) on the other. These genes are nearly identical, and in this plasmid they are controlled by the same promoter. Any variations in both colors indicates a global variation or extrinsic noise. Any variation in a single color over the other indicates noise particular to that gene, intrinsic noise.
As can be seen in the image, many cells are yellow (low intrinsic noise and high extrinsic noise), but some are more red or more green (high intrinsic noise and low extrinsic noise). The authors tune the level of expression and find noise increases as expression decreases. That is, as an input gets smaller the response is less fine tuned because proteins are made in discrete numbers. Additionally, notice how varied the response of an individual cell is to the average of the population. Even though these cells are growing right next to each other in the same media with the same inputs their protein expressions vary widely.