To get the cutting frame or Drill Spindle in the right position for cutting, it is held in a quick change tool post (QCTP) bolted to a Cross Slide.
The reason it is called a quick change tool post is because the ornamental turner can easily change the tooling. To do so, the lever (the ball-ended rod projecting off to the right in the picture) is rotated anti-clockwise and the tool removed. Then a new tool is added, and the lever rotated clockwise to tighten it down.
The most common type of quick change tool post is an Aloris style (shown on the right).They come in a number of sizes; and most manufacturers have tried to map to the Aloris sizes.
These can be purchased as either piston style and wedge style. The wedge format is recommended by some as they say it is less likely to vibrate loose.
A good source for the 0XA quick change tool post (and Cross Slide) is LittleMachineShop.com. A starter-level part is :
Fun fact : "Aloris" comes from the creator's family name, Sirola, spelled backward.