In our solar system, there is only one star that we know of - the sun! Our solar system is very unique in that is only has one star. Most other solar systems have at least two stars. These are called binary systems. Some solar systems with as many as six stars have been observed by astronomers.
Two paleontologists, David Raup and Jack Sepkoski, proposed in 1984 that there may be a second sun that is close enough to us to be seen every 32 million years (but still very far away!!), called Nemesis. They think Nemesis lies beyond what is called the Oort Cloud. The Oort Cloud is a giant cloud of icy space debris and comets that surrounds the sun.
The theory put forward by Raup and Sepkoski proposed that mass extinctions like the one that killed the dinosaurs occur every 32 million years - the same period of time in which Nemesis might be able to be seen from Earth. The most common theory for why the dinosaurs died is that a huge asteroid or comet (or a number of asteroids) collided with Earth. Theories propose that Nemesis may be disturbing the comets and space debris in the Oort Cloud, which sends this debris hurtling towards Earth.
In contrast, many geologists do not believe that mass extinctions happen so periodically: there is a lot of argument over whether these theories are correct! Research performed in 2010 and 2011 shows conflicting information: one study found that there was no evidence in the crater record for Nemesis; another study found evidence in the fossil record that extinction events may be periodic, but are twice as far apart as originally proposed by Raup and Sepkoski in 1984.
Astronomers believe that if Nemesis were to exist, it would have to be a red or brown dwarf. The search for Nemesis is complicated by the fact that our current technologies cannot easily tell the difference between a red or brown dwarf and a giant star from very far away. There are also thousands of different areas that could potentially contain stars that could be Nemesis, so the search is a huge one!
In 2012, nearly 2000 brown dwarfs have been identified in areas of space near our solar system, but not one of them has been found to be inside our solar system - so none of these stars are Nemesis! Two satellites tried to conduct astronomical surveys (one in the 1980s, and one from 1997-2001) to search for Nemesis, but both failed to detect an additional star in our solar system. Then in 2011, a senior scientist at NASA argued that new infrared sky surveys that have been undertaken in more recent years would have been able to detect Nemesis if it existed - but the infrared sky surveys have found nothing.
Nobody has proven that Nemesis exists yet, so at the moment, it remains purely theoretical. For now, we can only say one thing with certainty: that our solar system is unique in that it only has one star.