In cancer genomics experiments often involve sequencing a large number of tumour samples, but few or no matched normals. This is partly due to financial constraints, but also due to ethical ones: it is easy to get ethical approval for tumour tissue, which is removed as a matter of course in a patient's care anyway, whereas it is hard to get approval for normal tissue, which would not normally be removed and would require an additional procedure.
However, this presents a problem. Most of the procedures we use in standard gene expression analysis involve comparing one group to another after suitable normalisations of the data. But here we have no suitable comparison group. To get around this problem in cancer RNA-seq analysis we do something called outlier detection. That is, we assume that for a given gene, the majority of the patients have "normal" expression, and look for the few patients that don't look like the rest. We do this with the Z-score.
To calculate the Z-score for an observation we subtract the mean of all observations and divide by the standard deviation. Thus the Z score of an observation is how many standard deviations an observation is from the mean of all observations - or how unusual it is.
Thus for Gene A in Patient 1, we calculate how many standard deviations Gene A is from the mean of Gene A across all patients. A very big (or small/negative) value tells us that the expression of Gene A is unusual in patient 1 compared to the other patients.
(NB: in some cancer studies they use the median and median absolute deviation rather than the mean and standard deviation to compute a "robust Z-score" as the data is unlikely to be normally distributed).