Zero-space, or Z-space as it is usually called, is a dimension of white nothingness, where neither time, nor space exist. It can be passed through using a certain type of technology, allowing for instant faster-than-lightspeed communication via Z-space transponder and faster then light travel. Special Z-space engines are needed on a ship for it to travel through Z-space, though one could enter it via a black hole -- if one survives. Although the Andalites discovered Zero-space, the Yeerks quickly adapted it after stealing some of their ships in The Hork-Bajir Chronicles. Z-space is also important in morphing because that is where the excess matter goes when someone morphs a creature smaller than their regular body, or matter is retrieved from there when more mass is needed, such as a human morphing into an elephant. It was once thought that matter just clumped there in its basic elements, but in Book #18: The Decision, when the Animorphs morphed into mosquitoes, they discovered that the person, minus what was added or taken away for morphing, is kept in their regular form there.
It is important to note that Z-space is constantly in a state of flux, and so, Z-space routes that could take one day right now, may take take a month or ten years later to travel through. This is why the Andalite fleet, despite being informed of Earth's invasion, cannot arrive quickly. Also, there is an astronomically small probability that a ship flying through Z-space will come in contact with the mass of a being in morph. Generally, due to the velocity of the ship, the being will be killed. However, if caught in the wake of the ship, it is possible for the beings to survive. However, since Z-space is not conducive to life, a being would die unless rescued.