Perhaps you are referring to the situation where you do some type of analysis on high throughput data which results in a list of "interesting" genes. The typical scenario I'm thinking about is running some type of transcriptome analysis between two conditions of interest.
Let's further assume that this list of "interesting" (differentially expressed) genes has 100 members. Then what?
You want to ask if there is something "interesting" about this group of genes. I guess this is what "functional enrichment analysis" is referring to.
Do many of them seem to be involved in some type of pathway we know about? The Wnt signaling pathway, for instance? What if 10 of the genes in your list are in this pathway, is that more than you would expect by chance if you just fished out 100 genes at random?
To answer the question above, people typically do some gene ontology enrichment analysis.
You could also do something like GSEA, which doesn't require you to pick a cutoff that says this gene is interesting or not, but rather takes the entire list of genes in the experiment ranked by some metric (here, it would be the log-fold change between your two experimental conditions), then looks for enrichment of genes at the top of you list (or bottom) within different gene signatures that have been curated elsewhere.
If you google for "gene ontology analysis", or "gene set enrichment analysis" you'll get tons of information you can read through that will fill in the blanks of this answer, but this is the general idea.