If you rotate your camera around some randomly chosen point, your images will have parallax and your stitches will be bad.
With most lenses, there is one special point around which you can rotate your camera and get no parallax.
In the panorama photography community, that special "no-parallax point" is often called the "nodal point" or "entrance pupil".
Periodically there is debate about what it ought to be called. In optics, the term "nodal point" means something else entirely, a place in the optical path that is of no significance to most photographers. The term "entrance pupil" means the image of the limiting aperture, as seen through the front of the lens. "Entrance pupil" properly refers to an area, not a point, but the no-parallax point is at the center of the entrance pupil so these terms are closely related. The no-parallax point is also the "center of perspective", but this term is not commonly used and does not describe why the no-parallax point is important.
Everything considered, the best term for the no-parallax point may be (surprise!) the "no-parallax point". "Entrance pupil" is accurate although a bit imprecise; it makes a good term for searching the literature and may be preferred in formal writing. "Nodal point" is ambiguous and potentially confusing. It should be avoided when writing and interpreted with caution in reading the literature.
Regardless of what you call the no-parallax point, it is easily found by trial and error. Just adjust the rotation point so that foreground and background points stay lined up.
Some lenses, notably fisheyes, do not have a single no-parallax point. Instead, they have a range of what we might call "least-parallax points" that depend on the angle away from the lens axis. Such lenses can be recognized easily -- just look into the front of the lens and observe that the location of the entrance pupil moves forward or back as you rotate the lens off-axis. With such lenses, it is good to pick one angle at which you like to stitch, and rotate your camera around a point that gives no parallax at that angle.
To facilitate finding the no-parallax point for other people, please fill the measurements you have found for your Camera / Lens / Focal Length combination in the Entrance Pupil Database
Big Ben's Panorama Tutorials, Determining the Nodal Point of a Lens.
Alain Hamblenne, The Grid