Whenever light passes a boundary, it diffracts, or bends, due to the wavelike property of light interacting with that boundary. An aperture in an optical system, typically circular or circle-like, is one such boundary.
How light interacts with the aperture is described by the point spread function (PSF), or how much and to what degree a point source of light spreads as a result of passing through the optical system. The PSF is determined by the geometry of the system (including the shape and size of the aperture; the shape(s) of the lenses; etc.) and the wavelength of light passing through the optical system. The PSF is essentially the impulse response of the optical system to an impulse function, a point of light of some unit amount of energy that is infinitesimally narrow or tightly bounded in 2D space.
The convolution of light from the subject with the point spread function results in a produced image that appears more spread out than the original object. By Wikipedia user Default007, from Wikimedia Commons. Public Domain.
For a perfectly round aperture in a theoretical optically-perfect imaging system, the PSF function is described by an Airy disk, which is a bullseye-target-like pattern of concentric rings of alternating regions of constructive interference (where the light's waves interact constructively to "add up") and destructive interference (where the light's waves interact so as to cancel themselves out).
It's important to note that the Airy disk pattern is not a result of imperfect lens qualities, or errors in tolerances in manufacturing, etc. It is strictly a function of the shape and size of the aperture and the wavelength of light passing through it. Thus, the Airy disk is a sort of upper-bound on the quality of a single image that can be produced by the optical system1.
A point source of light passing through a round aperture will spread to produce an Airy disk pattern. By Sakurambo, from Wikimedia Commons. Public Domain.
When the aperture is sufficiently large, such that most of the light passing through the lens does not interact with the aperture edge, we say the image is no longer diffraction limited. Any non-perfect images produced at that point are not due to the diffraction of the light by the aperture edge. In real (non-ideal) imaging systems, these imperfections include (but limited to): noise (thermal, pattern, read, shot, etc.); quantization errors (which can be considered another form of noise); optical aberrations of the lens; calibration and alignment errors.
There are techniques to improve the images produced, such that the apparent optical quality of the imaging system is better than the Airy disk –limit. Image stacking techniques, such as lucky imaging, increase the apparent quality by stacking multiple (often hundreds) different images of the same subject together. While the Airy disk looks like a fuzzy set of concentric circles, it really represents a probability of where a point source of light entering the camera system will land on the imager. The resulting increase in quality produced by image stacking is due to increasing the statistical knowledge of the locations of the photons. That is, image stacking reduces the probabilistic uncertainty produced by diffraction of the light through the aperture as described by the PSF, by throwing a surplus of redundant information at the problem.
Regarding the relation in apparent size to brightness of the star or point source: a brighter source of light increases the intensity ("height") of the PSF, but does not increase its diameter. But increased light intensity coming into an imaging system means that more photons illuminate the boundary pixels of the region illuminated by the PSF. This is a form of "light blooming", or apparently "spilling" of light into neighboring pixels. This increases the apparent size of the star.