The question is not too broad, just involves a fair bit of work to both do the research and compose a response. I'll do the latter, but in brief.
The strings of the four nucleotides encode genes. Sometimes these genes are broken into protein-coding portions (exons) and sometimes intervening, non-coding regions (introns). Bacteria, for example do not have introns within genes while most human genes do. The nucleotides in the protein-coding portion of the gene do just that - give instructions on how to synthesize the protein by directing which tRNAs the ribosome uses to translate the genetic code into a protein chain.
A useful phrase is protein sequence dictates protein structure and that structure dictates function. So, the proteins do things and many act as enzymes or modulators of enzyme activity. An enzyme catalyzes a biochemical reaction, lowering the activation energy to go from reactant(s) to product(s). One enzyme can be thought of as one unit within a metabolic network. A modulator might activate or deactivate an existing enzyme. A protein kinase is a good example of a modulator.
Next, string some of those enzymes together to build a pathway. A common example is glycolysis. Think of this as a chain where the output or product of enzyme A serves as input to enzyme B, whose product is the input to enzyme C, and so on. These pathways are two-way, but it is exceedingly difficult to travel from the final product all the way up such a pathway to the initial input within the confines of a biological system. Thus, the pathways in practical terms are thought of as one-way.
However, the pathways do not function in isolation. The final product may serve as input to one, two or more other pathways. Your pathway of choice may begin with the output of another process. Furthermore, pathways can branch: Glycolysis can proceed to glyceraldehyde 3-phosphate and pyruvate, or directly after glucose 6-phosphate formation can be shunted to the pentose phosphate pathway to yield reducing power and 5-carbon sugars (to be used in ATP, CoA, FAD, NAD+, DNA and RNA). Now, pathways leading to other pathways and branches and such make a network.
You and your colleagues might want to know if genes within a metabolic network are over- or under-represented in a test compared to control. Or, genes for a given network are not present in one species compared to several others. Or, one could estimate the flux through such a network given a set of measures.
Peruse examples at both KEGG and Reactome, expecially at KEGG where you can view pathways pertinent to your organism(s) of choice.