During this time period in Greek history, the country that we now know as Greece was in no way a united country. Instead, it was made up of many small city-states. Each state was known as a polis, the plural of which is poleis. The poleis were not united with one another. Instead, each one could pursue its own interests. This led to, as your question suggests, chronic disunity. This disunity helped allow Xerxes to invade. Many of the poleis acknowledged the Persians as their overlords. Many of them helped to ensure that the Persians had the provisions they needed for their invasion. Thus, the invasion showed the chronic disunity of Greece.
However, the Greeks were also capable of banding together, at least in the face of an outside enemy. This is what happened when Xerxes invaded. The two most important poleis, Athens and Sparta, eventually formed an alliance and many other poleis joined as well. With their forces united, the Greeks were able to defeat the Persians and drive Xerxes out of Greece.
Thus, this invasion showed both the disunity of Greece and its ability to unite for a common cause.