How do periodic trends affect bonding?

The simple answer is the ionization energies and the electronegativities of the bonding atoms indicate how evenly or unevenly the atoms will share the electrons.

Atoms with high electronegativities and high ionization energies (non-metals) will hold onto electrons.

So if two non-metals bond, they will form a covalent bond.

If the electronegativities are very similar, the bond will be non-polar covalent.

If the electronegativities are different, the bond will be polar covalent, and the atom with the greater will be more negative.

Atoms with low electronegativities and low ionization energies (metals) can easily lose electrons, forming positive ions.

If a non-metal atom collides with a metal atom with sufficient energy, the non-metal may have enough electronegativity to pull the electron away from the metal atom.

This results in oppositely charged ions that attract each to form an ionic bond.

Here is a good figure to help you understand the trends at a glance: