If the earth disappeared in the morning, would anyone out there notice?
Obviously, the largest object in our solar system is the sun. All other objects - including planets, such as the earth, moons, meteorites, asteroids, etc. differ from the sun in two key aspects. First, they are much, much, much smaller. The sun itself contains >99.8% of the total mass of the solar system, that is, the sun is more than 500 times heavier than the mass of all other objects, combined.
The second key difference is that the sun is the only object in the solar system that emits light (shines). All other objects – including, of course, the moon - do not emit their own light, but only reflect the light that they receive from the sun. It is only at the sun that nuclear reactions take place, that produce the heat and light we get. Thus, by definition, the sun is the only star in our solar system.
The stars that we see in the night sky (when it is not cloudy) are all similar to our own sun in this basic respect: nuclear reactions take place within their cores, and as a result they emit the light that we see.
For centuries, philosophers and scientists assumed that extrasolar planets, namely planets that orbit stars other than our own sun, exist. However, clearly, as these planets are presumably much smaller than their parent star, and so neither do they emit light, then there was no way of confirming this hypothesis - which, interestingly enough is rooted very deeply into the (western) culture.
The development of stronger telescopes and detection techniques led, in 1994, to the first confirmed discovery of an extrasolar planet. Although this first detection took place only slightly over 20 years ago, by now (2015) nearly 2000 extrasolar planets have been detected, and this number is rapidly growing.
There are two main ways in which such detections are done, both are indirect. The first is to use the fact that when a planet, in its orbit moves in front of its parent star, it slightly "hides" it - similar to solar eclipse, that happens when the moon hides the sun. Of course, this cause a minute decrease in the light observed from the star, but current telescopes are strong enough to see this minute change.
The second technique uses the fact that the existence of planet slightly affects the motion of the star - not only the planet rotates around its star, but the star is also moving around the common centre of mass. This slight motion can be detected using the Doppler effect.
Both these detection techniques are very difficult to conduct; both are easiest when the planet is very close to its own star and very big. Indeed, the first planets that were detected were large, Jupiter-size planets that orbit very close to their star, in orbits which resembles that of mercury (the closest planet to our own sun).
However, as the telescopes gets stronger and stronger, and as the accuracy of measurements gets better, smaller planets that orbit further away from their stars are being detected. The "holy grail" in this field is to detect planets in the "habitable zone": these are planets that are both small enough and are sufficiently far away from their star that water could exist on them in liquid form - providing the necessary ingredient for life, as we know it. It seems that this goal was recently achieved, with the dedicated Kepler satellite announcing on July 2015 the detection of a planet called "Kepler 452b", which is the first planet which could potentially host life.
Thus, we are witnessing a very rapid growth in our knowledge of extrasolar planets in the past 20 years or so. I therefore have all reasons to believe that in the future - the near future, more and more extrasolar planets will be detected. Currently (2015), there are still no evidence for life in any of these planets. But of course, nobody knows whether life actually exists in any of these planets. If so, it is definitely a possibility that whoever lives there might be building their own telescopes, looking at us... in which case, they will be able to notice if the earth disappears one day.