Hi Alan. Great question! So H stands for hemagglutinin and N stands for neuraminidase. These are surface proteins on the surface of Influenza viruses. Other viruses also have these, but lets focus on influenza since that is what you're asking about.
These both have to do with the way influenza enters and exits cells. Influenza begins by targeting respiratory epithelial cells. How does it get in? Hemagglutinin forms a bond with sialic acid which allows it to become endocytosed into the cell.
Quick definitions again just incase:
Endocytosis - cell using its membrane or combining with an outside membrane to take in outside contents
Sialic acid - common carbohydrate residue on the outside of cell membranes for messaging and cell adhesion
When the virus has finished its replication inside the respiratory cells, it must leave in order to infect new cells. To do this, it has to cleave the bond with sialic acid it made in the first place. Neuraminidase achieves this. Tamiflu(oseltamivir) is the drug you take to treat the flu. This is an inhibitor of Neuraminidase!
Overarching summary - Hemagglutinin gets the virus into a cell; Neuraminidase gets it back out to infect new ones.
Extra info incase you need more in depth:
Antigenic Drift - Point mutations in the proteins that cause resistance to medications and potentially new strains resistant to old antibodies.
Antigenic Shift - This is more complex. Orthomyxoviruses(fancy word for the family that the flu viruses are a part of) are segmented. This means that their genome(negative sense ssRNA: -ssRNA) is cut into different parts, specifically 8 different parts. There are many different types of H's and N's. There is the currently circulating H5N1 avian flu virus, as well as H1N1 and H7N9. Lets say that both of these strains(H1N1 and H7N9) infect one cell. They can mix and match their genomes! You can now create H1N9 and H7N1 just from both infecting one person. These are the major things that cause influenza pandemics.
I Hope this helps! If you need more specific details on protein structure or more in depth functions please feel free to ask :)