This question is puzzling, and posed one of the most controversial physics questions in the twentieth century.

Let's look at the problem through Newtonian mechanics. The idea of a second sun in our solar system is very outrageous, so there are either two ways it could work. The first is that there is an incredibly small star orbiting our Sun, but is so dense, and so close to the Sun, we can't see it. This potentially could work if the star was heavy enough to counteract the sun's mass, and if it's distance away from the sun was small enough that the two-star system could effectively turn into the one star system. We want it to look like a one-star system because that is the system our planets' orbits follow. This was hypothesized by many physicists in explaining the strange orbit of the planet Mercury in our solar system. The star was named Vulcan, but, due to Einstein's relativity theories and recent evidence, was found not to exist.

The next way this system could work is if there exists, an incredibly far distance away from us, some star orbiting our own sun. This could be true if the star were far enough away at all times that it had no considerable effect on the planets motion. The star would act as a sort of "super" comet, crashing into things and pulling with it any smaller bodies trapped by its gravity. This is a theory made to explain mass extinctions that have occurred on Earth every 26 million years. Similar to how the dinosaurs supposedly went extinct, scientists say this star comes barrelling in every 26 million years, and with it, bringing mass destruction.

This is most likely as false as the above claim, however, the fun part is we won't know the answer until we see it happen with our own eyes!