GO = gene ontology. The "meaning" of the gene (biological function, molecule type, cell location, disease association, etc).
Gene enrichment analysis is a search for patterns in gene expression, gene upregulation or downregulation, in lists of total significant genes, etc. Most genes are associated with some pathways, for example:
ACTB (beta actin) and cell migration and cell proliferation and mitosis,
TP63 (p53) and cell cycle arrest and apoptosis,
ALDH2 and aldehyde metabolism,
CDC7 and cell cycle control,
CDK2 and cell cycle control,
CDK7 and cell cycle control and transcription, etc.
The analysis looks at the specific genes and their associated biological pathways or molecular functions or disease associations or upstream regulators, then calculates p-values for how often those associations come up compared to how often you would expect them to come up in a random list of genes. If an association comes up really often (like cell cycle control), is it because it has a large list of associated genes since it is a well-studied pathway, or is it because your specific list of genes is "enriched" for that pathway? The analysis helps you determine that.
The most common statistical test for this type of analysis is a Fisher's Exact Test, which calculates the significance of an overlap. The overlap you are interested in is [your genes] compared to [the genes associated with a pathway]. The length of genes in either list is taken into account, since the likelihood of an overlap is higher with longer lists. You calculate a p-value for every pathway to determine which pathways are significant for your gene list.