Convergence and divergence roughly determine where air will be sinking or rising. Rising air is associated with clouds and precipitation; sinking air is associated with clear, calm conditions and good weather.
Why do I care? Convergence and divergence make the difference between a grey day and a sunny day.
I should already be familiar with: Pressure, What Drives Weather
Figure A: Rising/Sinking Air Within Pressure Systems
At the surface, low pressure systems have a counterclockwise rotation in the Northern Hemisphere, with the wind turning slightly inward towards the lowest pressure. This causes air to converge, or come together, at the center of the low near the ground. Since the converging air has nowhere else to go, it rises. As the air rises, the water vapor within cools and eventually condenses into cloud droplets and raindrops. Because of this, low pressure centers are generally associated with clouds, precipitation, and what we generally call "bad weather."
Surface high pressure systems in the Northern Hemisphere have a clockwise rotation with wind turning slightly outward away from the highest pressure. This causes air to diverge, or move away, from the center of the high near the ground. Air has to come from above to fill in the void left by the diverging air at the surface. This leads to sinking motion in the center of a high pressure system and generally clear air. This explains why high pressure systems are associated with what we think of as "good weather."