I often hear about planetary ring systems, and even some moons might have them, but how about stars? Can a star also have rings?
They certainly can. A ring is often formed around a celestial body when its gravity rips apart another smaller celestial body. The Sun is really massive, so it could destroy any object that is not dense enough. Just Google about the Roche Limit for more informations (and better explanations).
Now, take a look at our solar system : You have the asteroid belt between Mars and Jupiter, and you also have the Kuiper belt beyond Neptune. Would you consider these being rings ? They sure are not as smooth as Saturn's, but to my mind, they still are rings.
The first star having rings seems to have been discovered in the late '90 by You-Hua Chu (University of Illinois). In Cosmic Catastrophes: Exploding Stars, Black Holes, and Mapping the Universe, Craig Wheeler wrote (in 2007):
[You-Hua Chu] scored another coup a decade later, at a meeting in Chile to celebrate the tenth anniversary of the discovery of the supernova, when she reported that she had discovered the first star to have rings around it, like the progenitor of SN 1987A.
SN 1987A was a supernovae that exploded in 1987. The progenitor (the star that actually exploded), Sk-69 20, was a blue supergiant. Pictures taken from Hubble showed three rings:
This video shows the lightning effect of the explosion on the inner ring from 1994 to 2016, that is, 7 to 29 years after the explosion (the ring is a few light years across while the most rapidly moving outer portions of the exploded star are moving at least at 10 percent of the speed of light - source: same book, p. 136).