The word "Emanation" comes from the Latin e-manare, "to flow forth". The cosmos and finite beings are all seen as having emerged out of the Absolute Reality through a sort of "out-flowing". Metaphors are with the ocean (the Absolute) and the waves (the Universe); the Sun (the Absolute) and the Light that shines from it (the Universe); a fountain (the Absolute) which overflows (the universe); and so on.
diagram of bands of progressively denser matter projecting into the Universe.
Gouache on cloth
According to Emanationism, Creation occurs by a process of emanation - "out-flowing". The entire cosmos, and even all the Gods and Godheads beyond the Cosmos, has come about through emanation. Just as the ocean forms its surface into waves, so the Absolute forms upon and as Itself successive manifestations, successive entities. And these in turn create - or rather, emanate - further entities, and so on, with all these entities combining and interacting in the extraordinary network of existence.
Each of the levels of reality in the Emanationist Cosmology could be termed a "World". Here, "World" is a general term meaning any self-contained realm or universe of existence. One could equally well say "Universe", "Cosmos", "Sphere", "Realm", "Plane", "horizon", "reality", "state of existence", "state of consciousness", etc. The term "World" has been chosen simply because it is a useful general term.
One could think of the relationship between each of these levels as being like "body and soul", "spirit and matter", or "Creator and creature" , in that each higher level is the Soul, Spirit, and Creator of the level immediately below it; and the Body, Matter, and Creature (created being) of the level immediately above it.
diagram of the relationship between spirit and matter, according to the emanationist paradigm
Each of the levels or stages in this "spectrum" or "great chain of being" has its own specific characteristics. So you could speak of the psychic world (or "astral plane"), the angelic world, the archangelic world, the Divine world, and so on; hierarchy upon hierarchy, world upon world, a kind of epiphany or manifestation of the Divine; all looking downwards to matter, and also looking upwards to the godhead.
Such a Cosmology has to be lived. It cannot be a mere theoretical thing. It must be an actual Vision of Reality. The Vision of Worlds beyond Worlds, of hierarchies of Angels or Gods arranged in order upon order, Light above Light, is a truly magnificent and awe-inspiring one. It is not abstract speculation. It is contemplative reality.
 see e.g. L.I. Krakovsky, Kabbalah - The Light of Redemption, pp 19, 86-7, 125 (Research Centre of Kabbalah, Jerusalem, 1970)]
Emanationism in contrast to other theories
The Ontodynamics of Emanation - An Exercise in Speculative Metaphysical
Emanation and Ascent in Hermetic Kabbalah 1.4 Mb PDF - informative presentation and notes by Colin Low on emanation and the roots of Hermetic Kabbalah
Emanationism - wikipedia
Emanation - Internet Encyclopedia of Philosophy - comparative overview