Not organized along feudal lines (as the rest of the universe is), the Tleilaxu are secretive, and very little is known about them except for their biological products (replacement parts and gholas), and their face dancers. No one has ever seen a Tleilaxu female.
The use of "Bene" before their name suggests that they are an order of some kind, like the Bene Gesserit; this becomes clear in the last two books of the series.
A ghola is a clone, grown in an axolotl tank from the DNA of a another person. At the time of Dune, gholas have no access to the lives of the person from whom they were cloned, but after Dune Messiah, it is discovered that a ghola can recover his or her genetic memory during a carefully staged moment of great stress. Much later in the series, it is revealed that the masters of the Tleilaxu have been using this as an improvised device for immortality: at their death, they are cloned; their clone recovers its memories, and the masters, in their serial incarnations, have memories stretching back thousands of years.
An axolotl tank is essentially a braindead woman whose womb is used as a tank to create gholas and other creatures. In Heretics of Dune, it is revealed that the Tleilaxu can also create the spice melange in axolotl tanks, breaking the monopoly on spice that Arrakis held for thousands of years which strongly determined the economics and the politics of the Imperium.
A face dancer is a mimic. Face dancers are sterile creatures, with full sentience, but with a genetically programmed loyalty to the Tleilaxu masters. They can be explicitly controlled by being forced into an hypnotic state with some predefined sound (often a specific whistling noise). Face dancers are used by the Tleilaxu throughout the universe to impersonate key people after killing the originals. Only the Bene Gesserit are able to detect a face dancer.
Over the course of the series, the Tleilaxu try to create more perfect mimics, to the point where the face dancers lose their ability to mimic and their awareness of themselves as face dancers, so perfectly do they mimic their subjects. At that point, the face dancer effectively becomes the person he or she is mimicking, and passes beyond the control of the Tleilaxu.
In the last two books, the core of the Tleilaxu is revealed: they are the descendants of a Muslim sect, and are organized along theistic lines. They have spent thousands of years concealing this fact, waiting for their ascendancy, which they believe to be occurring in Heretics of Dune.