I am not an expert but am happy to share what I know that answers some of your questions.
-What does it mean to work wireless with a flash?
(A) Simply that an off-camera flash is controlled by the camera but not connected by wire. The control may be by pre-flashes from the built-in to an off-camera flash. Or it can be via radio signal by using add-on controller/receivers to camera and flash.
-Is it even possible to work wireless with just one flash
(A) Yes, I do all the time. My K20D built-in flash that starts when it flashes, starts the slave flash also. The slave must be able to see the first flash in order to function. I can have the slave flash on a bracket only 20cm from the camera body for macro shots when using both flashes, or have the slave flash on the other side of the room for portrait lighting.
-With which flashes is it possible to work wireless?
(A) I can't give you a complete list. The Guide you mentioned is a good start if you are looking at Pentax compatible wireless without buying radio signal gear.
-What do I need to work wireless?
(A) To start, only your K20D set in wireless flash mode and a Pentax compatible (see the Guide) off-camera flash set in slave mode.
P-TTL is simply the Pentax version of TTL (Through The Lens). I have never researched the technical differences. I will have to leave that for you to look up and post for me to learn.
The Guide is probably most helpful as a shopping guide once you understand the terminology and technology. Come back to it in a week. As far as the terminology, Wikepedia and Google are your friends as well as old threads in this Forum.