I don't understand what differential expression is, and an article I read mentioned it. I can't find anything to actually answer my question. Can anyone help me better understand what this means?

The link to the article is: https://tcr.amegroups.com/article/view/957
Very basic summary:

Let's say that you have two samples from the same patient. One sample is from a kidney tumor biopsy. The other sample is a biopsy from the patient's other kidney, which seems to be perfectly healthy tissue.

Theoretically, you would expect that the two samples will have different amounts of certain messenger RNA transcripts, because the cells within the two tissue samples are performing different functions and are thus transcribing different genes. **You're curious to see which transcripts from the tumor are being produced at a significantly higher or lower number in the tumor tissue than the healthy tissue. These differences in transcript production in aggregate comprise the differential expression that you're studying.**

For modern differential expression studies, you're going to make sequencing libraries from the messenger RNA transcripts produced by the cells in each sample. You'll sequence those transcripts and then (after some cleanup of the raw sequencer data) align your sequenced reads to the genome. You'll see which regions of the genome have coverage, and compare the depth of coverage for the regions between samples to determine which regions have significantly different coverage depth. Then you can compare the transcripts of interest against a database of known transcripts to determine the genes (and the isoforms of the genes) that are being differentially expressed.

Now, in a real study you're going to ideally have multiple patients and tissue samples from several sites on each patient. You're not necessarily going to be studying tumors, either. There are lots of potential transcript expression patterns to be gleaned for pretty much any biological phenomenon through careful study design.