The Jewish calendar, unlike the secular/Gregorian calendar, is a lunar calendar.
The first commandment given to the Israelites in Egypt (Exodus 12) is to count and begin the months with the new moon (Rosh Chodesh in Hebrew). One ancient custom is to recite kiddush levana - a sanctification of the newness of the moon - on the first Saturday night in the new month when the moon is visible. Kiddush levana is recited outdoors, standing underneath the open sky.
While sanctification is generally accomplished with just a blessing, there is quite an extensive text that is recited as part of the kiddush levana ceremony. One of the strangest rituals of this ceremony is the custom to greet three individuals by saying “Shalom aleikhem” - Peace unto you. They respond “Aleikhem shalom” - peace is upon you. Why?
1. Having greeted God in the original blessing, we wish the blessing of peace upon each other. Our religion emphasizes that our relationship with God must not be at the expense of our relations with fellow humans.
2. According to the Midrash (Biblical legend), the sun and the moon were created to be the same size. However, the moon "wished" to dominate the heavens and suggested to God that He designate the moon as the primary authority of the heavens. God punished the moon for its haughtiness, diminishing its size. The sun nevertheless shares its light with the moon, and thus teaches us not to bear a grudge against those who may have wronged us. Wishing peace upon others is a way to express this idea this sense of forgiveness.