As far as I've seen, this expression is almost always used in relation to gene expression profiling. Unfortunately, I have no background in this area. Can someone please explain this in layman terms?
Although each cell of your body essentially contains the same DNA and the same genes, cells in different tissues express (turn on) different genes under different conditions. Measuring differential gene expression involves looking at the amount of expression for a gene (or set of genes) in two contrasting scenarios. The contrast could be across different times, different tissues, different conditions, different related species, etc.
When, you say a gene is "differentially expressed", this is very context-specific. The phrase means nothing by itself, and it is only useful in terms of the applicable contrast. For example, the statement "gene A is differentially expressed" is uninformative, while the statement "gene A is differentially expressed in liver and muscle tissue" is descriptive--it tells you that liver tissues and muscle tissues have a significantly different level of gene A products. Often the terms "up-regulated" and "down-regulated" are also used to provide additional detail. In the context of the previous example, the statement "gene A is up-regulated in muscle tissues" tells you that the level of gene A products is higher in muscle tissues than in liver tissues.