Short answer
Kabbalah teaches that reality unfolds through four worlds, or levels, descending from the most spiritual to the physical: Atzilut (emanation), Beriah (creation), Yetzirah (formation) and Assiah (action). They are often presented as stages between the infinite (Ein Sof) and our everyday world.
The scriptural hook
The four names are traditionally linked to a verse in Isaiah (43:7), which speaks of creating, forming and making. Kabbalists read these verbs, together with the idea of “emanation,” as pointing to a four-stage structure. The scheme is developed in the medieval Kabbalistic literature, including the Zohar, and was expanded in later systems.
Atzilut: the world of emanation
Atzilut is the highest of the four worlds. It is described as a realm of unity so close to the divine that there is little sense of separate existence. Kabbalists associate it with the purest level of the sefirot. Many teachers say it is a world where light has not yet become separate from its Source.
Beriah: the world of creation
In Beriah, the first sense of “something” distinct from God emerges. It is associated with the throne of glory in some sources and with high-level intellect. Sources link this world with the sefirah Binah.
Yetzirah: the world of formation
Yetzirah is where forms and emotions take shape. It is traditionally the world of the angels in many accounts, and corresponds to the emotional sefirot, from Chesed through Yesod.
Assiah: the world of action
Assiah is the world of action, which includes the physical universe. It is where moral choices are made, and for that reason Kabbalists give it high importance despite its being the “lowest.” Many teach that the aim is to bring holiness into this world.
Four or five worlds?
You will sometimes see five worlds. The additional world is Adam Kadmon, the “primordial human,” a level above Atzilut that features in the Lurianic system. Some modern teachers, including those whose course material is written for broad audiences, speak of five worlds for this reason. Others retain four and treat Adam Kadmon as a stage before the four. Both approaches reflect real traditions, and the difference is mostly one of counting. See Lurianic Kabbalah.
The worlds and the soul
Kabbalistic psychology links the worlds to levels of the soul, usually named nefesh, ruach, neshamah, chayah and yechidah. The lower levels correspond to Assiah and Yetzirah, and the higher ones to Beriah and beyond. The worlds are also linked to the structure of prayer, where the morning service is sometimes explained as an ascent through these worlds.
The worlds and the sefirot
There are different ways of combining the two frameworks. A common approach says that each world contains its own full set of ten sefirot, so the Tree of Life is a pattern repeated at every level. Another gives each world a dominant sefirah or group. Because this is complicated, treat any simple chart as a summary, not a complete account.
For the wider context, see Kabbalah Explained and Kabbalah Spiritual Practices and Meditation.