I was going through my Social Media Feed and found the attached post too frequent. The caption reads this is the best image of our Sun. Just as an example, the Universe Today's This is the Highest Resolution Image Ever Taken of the Surface of the Sun
Why? What exactly are the black lines that appear to be like kind of a web, and will such patterns be observed if in case the star was not the Sun but some other star? Are they thought to be common?
The dark lines are colder areas at the edge of the convection cells, where the cooled down plasma sinks towards the inside of the Sun. Now "colder" for the surface of the Sun, is still pretty hot, as explained here.
The yellow parts are where the plasma rises to the surface. Each yellow spot (which is actually the size of a country) is called a granule, and this web-like appearance is called granulation.
In the outer part of the Sun (the convection zone in the image below), there is convection, that is hotter plasma floats towards the top, cools down at the surface, and sinks back down, like in a lavalamp.
The existence of a convective zone in the outer part of the star is determined by the mass of the star, and all stars with a convective zone in their upper layer are thought to have such granulation patterns. So stars like our sun, or smaller have these patterns.
For larger stars, though, the convective zone is in the inner part of the star, and the outer part of the star is the radiative zone, so there might not be the same patterns on the surface.
Usernumber's explanation of the light and dark regions is correct, but there is more detail to be added about granulation on other stars.
Granulation is expected on other stars with surface convection zones, but the properties and timescales of the granulation can be quite different.
On the Sun, the granules appear and disappear in timescales of 10-30 minutes and the granules have a characteristic diameter of around 1500 km. There are thus about 4 million of these visible on the solar photosphere.
The size of the granules is expected to vary as the gravitational scale height in the photosphere, which is proportional to $T_{\rm eff}/g$. Thus stars with lower temperatures (K- and M-stars) are expected to have smaller granules, but stars with lower surface gravities (subgiants and giants) are expected to have much bigger granulation patterns (Cranmer et al. 2014).
In fact, given that gravity scales as $R^{-2}$, the ratio of the radius of the star to the size of a granule gets smaller as gravity decreases. Thus giants are expected have far fewer, but bigger granules.
The timescales are also different. The frequency of granulation appears to scale with the peak frequency of p-mode oscillations, which in turn scales as $g/\sqrt{T_{\rm eff}}$, and so cooler stars have higher frequency granulation, but giants, with 1-2 orders of magnitude lower surface gravity have much more slowly changing granulation patterns (Kallinger et al. 2014).
The truth of the above has been basically confirmed using disk-integrated variability seen in stars monitored by the Kepler satellite.
Of course, the granulation pattern cannot be imaged in distant stars, except in those stars with the largest radii and largest granulation patterns. There have been claims that surface brightness variations on Betelgeuse are due to granulation, but the first really believable images are of the close hypergiant $\pi^1$ Gruis (Paladini et al. 2017). This star is half the temperature of the Sun and it's gravity is about $10^5$ times lower. According to the ideas above, the granules should be 50,000 times bigger than on the Sun, i.e. a diameter of 75 million km.
The radius of $\pi^1$ Gru is about 250 million km, so its surface will be covered by only around 100 granules, roughly in agreement with what is observed (see below).
VLT near infrared image of $\pi^1$ Gru (ESO).
I'll add to @usernumber's answer some graphics. Unfortunately we can't yet "has YouTubes" for some reason so I'll just add the links.
There are two videos of the Sun linked in Phil Plait's Bad Astronomy article
Here are the same kind of convection cells shown in more familiar settings:
