Earth only has one Sun, but if in the beginning of our solar system there were two Suns, well that would have changed everything. Most likely if that was the case we would not be discussing this right now, Also our Earth would not even exist as we know it. The whole solar system would look a lot different since in the beginning phase when the stars spew out material we get planets out of that material, but if we have two stars then obviously we would get completely different planets and even then most of these planets would likely be thrown out of the orbit since the orbit would be unstable due to the two stars being so close. But still there are actually planets with two stars that are around us in our Milky Way galaxy itself.
One of them is in the Kepler-34 system, in this system there are two stars that are similar in size to our Sun and because of that they orbit a center where the center of mass is at. Around these stars orbit two planets of about Saturn’s size namely Kepler-34a and Kepler-34b, so with that system we get a pretty good picture of how our solar system would look like if it had two stars. If our solar system in the beginning had two Suns then there probably be just a few planets and rest of the stuff would probably be gone out of the orbit sent flying into space, so this is the scenario in which there were two Suns in the beginning of our solar system.
But what would happen if suddenly a new star appeared similar in the mass to Sun and close to Sun? Well, what would happen is a catastrophe, so our Sun and the new star that just appeared would then form a center around which they will orbit but rest of the planets would be ejected out into the outer space away from the Sun and that is because the point of center of mass was changed, what that would mean for us here on Earth is that, if suddenly a new star appeared it would be only cool for a little while as there would be day on the other side of the Earth and obviously there would be two Suns making the scenery overly bright.
But as the days would pass, we would be noticeably further away from the two stars and as the Earth would be getting further away the temperature would then drastically drop. From the poles the ice formation would start to spread, humans would obviously be in panic as there would be nothing they could do to stop the sudden freezing of the Earth, even nuking the Earth would not help much in this freezing event. As the Earth would be moving further away from the two stars, it would result in a surface of Earth being completely pitch black at some point as there would be very little or no light reaching the Earth from the two stars.
In this scenario most of the life on Earth including humans would not survive, the only ones that would get to survive would need to live under the oceans where there is liquid water but they would also need to have enough power to survive on the water but the life they would live would not be so good. So, even if a second star were to suddenly appear in the solar system that would be horrible news as it would nearly freeze us to death. So, that was the scenario in which Earth is ejected, now let’s see what would happen in a scenario where the Earth is in orbit around 2 Suns without much of a problem.
In a scenario where the Earth has 2 Suns and is comfortably orbiting around them in the same position as before then things would at first be very interesting, obviously the day in the day-night cycle would be longer depending on how apart the 2 Suns are but most importantly the luminosity of the two stars combined would be enormous which would be double the current luminosity and what that would mean is that more sunlight would be reaching the surface of the Earth which would cause drastic temperature rise and that would be immediately noticeable.
Snow would soon start to disappear and as the temperature would rise, it would have a sort of effect where suddenly the surface would be trapping more heat than releasing causing the temperatures to go higher and higher and it can go up to the point where oceans would dry up, off course in this scenario no life would survive since one of the most essential ingredient for life is water. Even if few people could survive if they somehow escape into space with water but if that were to happen they would still not survive for much longer because we still cannot infinitely survive in space without help coming from the Earth.