Why is hydrogen bonding and polarity important?
Both of these result in fairly strong intermolecular forces between different molecules. This has important applications to solubility for example.
Polarity in a molecule results in charges on opposite ends of that molecule that Will cause these molecules to interact quite strongly and that can lead to significant reductions in potential energy in these interactions.
One example is in solubility, where dipole-dipole interactions among molecules is the main contributor to a stable solution.
Hydrogen bonding is another type of dipole force seen particularly between H atoms that are bonded to highly electronegative atoms like F, O and N. These bonds are essential in the structure of proteins and DNA, both in terms of creating the molecules, and in determining there three-dimensional structure.
There are many other applications (too many to try to list here!) but all come down to the fact that these two features produce relatively strong electric interactions between molecules. Not as strong as ionic bonding, but stronger than dispersion forces like van der Waals forces.