One symbolizes unity, agreement, simplicity. When something exists alone, nothing disturbs it. It remains completely at peace, without regard for anything else.
Two symbolizes duality, tension and complexity.
The number three symbolizes a harmony that includes and synthesizes two opposites. The unity symbolized by the number three isn’t accomplished by getting rid of number two, the entity that caused the discord, and reverting to the unity symbolized by number one. Rather, three merges the two to create a new entity, one that harmoniously includes both opposites. On its own, Entity A leaves no room for Entity B, and Entity B does not allow for the existence of Entity A. Entity C demonstrates how A and B really are compatible, and even complementary. Bringing together two opposites (A and B) requires the introduction of an entity or common goal (C) that is greater than both of them.
The first three days of creation serve as an example for these three stages. On the first day, life had not yet been created. At the conclusion of the first day, the Torah writes, “It was evening and it was morning, one day.” Our sages comment on this verse that it was a “day of One.” G‑d was the only existence; even the angels had yet to be created. There was complete and simple unity.
The second day of creation symbolizes separation and discord. On that day G‑d separated the waters above the heavens from the waters below the heavens. This generated the first dichotomy. Now there were two opposites—heaven and earth, spirituality and physicality. Unlike the other five days of creation, regarding the second day the Torah doesn’t say that “G‑d saw that it was good.” When there is conflict and tension, the situation can not be qualified as “good.”
On the third day G‑d confined the waters to the seas and uncovered the dry land, which allowed for the creation of vegetation later that day. Now the light, the dark, the heaven, the earth and the waters were all working together harmoniously to produce. On the third day the Torah tells us that it was good twice. The third day didn’t undo the separation of the first two days. The polarity still exists, but the polarity became part of a greater unity. And that is doubly good.
When the two opposites merge and work together, the resulting unity is greater than the unity of one being on its own. Because the number one symbolizes peace that is achieved by excluding all others.
In a nutshell, one stands for unity. Two stands for disagreement. Three stands for the harmony of opposites that includes the properties of one and two. This idea, of the three stages, is especially common in the works of the Lubavitcher Rebbe. For examples of this concept, I recommend you read:
The Mathematics of Marriage, which demonstrates these three levels in marriage, our relationship with G‑d and the giving of the Torah.
The Third Month Family, which demonstrates that the “three model” of unity is the key to successful parenting.
Rabbi Eliezer Posner