I cannot find a clear explanation of what is mRNA expression level, and how to measure it. I would appreciate if someone explained it or gave a reference.
The RNA expression level is an estimate of the amount or proportion a given gene's RNA found in a sample of cells (such as a tissue sample) or even a single cell.
Expression is used because the RNA sequence is translated from DNA to RNA according to the signals within the cells.
There are various means of estimating this. They often involve a sequence specific probe - a short length of DNA usually - which will bind specifically to the RNA in question. The exclusiveness of the probe sequence, which is usually 20-45 bases long, to the RNA in question can vary and so this is a bit of an art sometimes.
Another category of expression estimation include RNASeq - which will sequence all the RNA polymers in a sample and estimate all the sequences at one time. The result is similar to a microarray, but RNASeq does not necessarily require the use of specific probes to an RNA to measure it, though it can.
These estimates are usually relative - except for careful qPCR perhaps. The numbers are not usually interpreted as "X copies of the RNA" in the sample.