A morphing cube (also known as an "escafil device") is an Andalite device that when operated gives the user the ability to 'acquire' a copy of DNA patterns from any animal that physical contact is made with, allowing the user to transform into that animal through morphing.
The morphing technology is not a single machine, but a vast network of thousands of microbionic units. These units are contained in the Escafil device, which is the central computer for the smaller morphing units. However once the morphing ability is acquired, the Escafil device is no longer needed for it to function.
It is also important to note that the Escafil device and the morphing power involve two different yet interconnected processes: One is the conversion of one animal's DNA into another. To morph, for example, a wolf, every cell in the Animorphs' body must convert to a cell of a wolf. The other involves simply rearranging matter in Z-space.
After the body is converted to another organism's DNA, it still retains the properties of the real person, which are preserved in Z-space. In fact, Z-space is where the mind goes, which makes sense because many morphs are too small to hold a human brain and all the cells it takes. The thought-speak ability gained by the Animorphs when they are in morph is because of this remote connection. These properties gradually deteriorate, starting from the time a person is in morph. Over the minutes, the real person in Z-space looses their original characteristics of human, or Andalite, which ever the case may be. By two hours, the process has gone so far that it cannot be reversed by morphing. Morphing has become impossible. The mind-link is still there, though, or else the "nothlit" would find themselves trapped in Z-space and their former body an ordinary wild animal back in real-space.
What actually accomplishes the morphing process involves microscopic agents that are absorbed into the body through the skin by touching the Escafil device, and the DNA of the animal is also acquired by touch. It is some sort of pseudo-virus which is sophisticated enough to be on the morpher's skin and, when it makes contact with another animal and the thought-controlled "acquire" sequence is initiated, takes a sample of the creature's DNA (which can be taken from a single cell, any cell) and relays the code to all the other units in the morpher's body.
Various clues and references have lead us to believe that they are located in the blood. Whether they are floating around in there alongside the blood cells or may actually be bonded with the morpher's DNA. This is not likely though, since the morphing power does not have an effect on one's genetics, as we can see from the Andalites. If it did, the first Andalite generation to have the morphing power would differ sligthly from their children.
The morphing machines are thought-controlled. One must concentrate and think about the animal one wants to morph before being able to do it, strong emotions can interfere with the morphing, and having a love for and being very conscious of the animals gives one expertise in morphing. The morphing units are referred to as pseudo-viruses because viruses exhibit all the qualities of life except for one: they can't reproduce on their own. They have to use a host cell to do the job. Since a virus is nothing but a protein shell with genetic material inside, to transform a cell into a virus-making factory, they latch onto a cell, and inject their genetic material inside (DNA or RNA, definitely DNA in this case).
This DNA has the programming to "hi-jack" the cell's nucleus and re-program it to make viruses. After a while, either the cell explodes from all the viruses, who go on to invade other cells, or the virus's DNA bonds with the cell's own DNA and remains dormant until something activates it and it becomes active again, turning the cell into a virus-making factory. A similar process could be used to change the morpher's native cells into the cells of whatever creature they're morphing. This means that for every cell of every species the morpher is capable of morphing, there is a pseudo-virus with the correct genetic material to make it.
There is a central unit for each creature the morpher has acquired, a cell-sized mechanism. In this mechanism are every kind of pseudo-virus needed to make the new animal. Not long after developing the morphing technology, the Andalites made contact with the Arns and used their biology knowledge to create the quantum virus, which also have microscopic units and also have Z-space relations. The quantum virus is the next step up from the morphing technology, except morphing units are constructive, while the quantum virus is destructive.