The realm known as Nekudim is one of the many worlds described by Kabbalah as being part of the order of development that G-d utilized to create the physical world.
Nekudim is the second world to result from Adam Kadmon
The source of the world of Nekudim
In this world, light emanates out of the "eyes" of Ak, and descends to encompass Ak from "naval" to "feet."
"Lights" and "Vessels" in the world of Nekudim
In the realm of Nekudim, ten individual vessels are created to contain the ten lights of this world. The vessels are small, undeveloped or "immature." Relative to the dully developed vessels that will come to exist in the world of Atzilut
, the vessels of Nekudim are merely "points." Therefore, this world is called Olam Ha'Nekudim, "the world of points."
Unlike the world of Akudim, where the ten lights flow freely in and out of the single vessel in the process of mati v'lo mati , in the world of Nekudim all of the lights enter, forcefully, into their respective vessels and break them.
The world of Nekudim is generally referred to as olam Hatohu, the world of "chaos." Often the two world of Akudim and Nekudim are considered to be two formsof "chaos," relatively "stable chaos" (Akudim, the form of "chaos" which doesn't break) and "unstable chaos" (Nekudim, the form of "chaos" which breaks). In modern "Chaos Theory" these two forms of "chaos" are described in relation to the physical world.
Manifestations of the world of Nekudim in human beings
In the human soul, this can be compared to a youth who possesses distinct but limited language skills, which hold him back from truly expressing his emotions.
In the human soul the shattering of vessels of Nekudim is like a psychological "breakdown" (resulting from the trauma of adolescence). Relative to the physiology of the human body this is like "death." The lights ("soul") returns to its source, the broken vessels ("the physical cadaver") falls into the lower realms of reality ("burial ground"), but contained within the broken vessels are "sparks" of life-force which, though temporarily hidden insure the eventual "resurrection" of the broken vessels.
The Emissaries who correspond to Nekudim
The lights of the world of Nekudim, which emanate from the "eyes" of Ak, correspond to the spiritual soul-root of those souls of Israel whose primary goal and purpose in life is to act as a "messenger" or "emissary" of God and His Mashiach ("Anointed One") and bring the light and goodness of God in order to permeate all consciousness with the reality that "God is King of the Earth."
According to this explanation, the meaning of the "breaking of the vessels" is that these first "messengers" did not succeed in their mission. Rather than recognizing that God alone rules over all of reality, each of these "kings" of tohu (hinted to in the Torah as the kings of Edom that rules before the kingdom of Israel) said "I shall rule," as taught at length in Kabbalah.