Arthur C. Clarke has a fantastic quote about alien life:
Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.
Clarke is correct that both thoughts are scary. The universe is huge and quite possibly infinite. To say that Earth is the only celestial body that harbors life means that the universe is a very lonely place. On the other hand, the possibility of an alien species more capable than our own is also scary.
In terms of alien life, a lot of people immediately jump to the conclusion that you or I are talking about humanoid aliens. That I think is unlikely. There are millions of species of living things on Earth, and most of them are not humanoid. Additionally, a great many species of living things on Earth are much more capable of living in extreme conditions than humans. Archaebacteria are a good example. They are often called extremophiles because they thrive in places that no other life can exist in. They survive incredibly hot temperatures at extreme ocean depths by volcanic vents. They survive extremely salty environments, and they are capable of living in oxygen free environments as well. Some bacteria are chemotrophic autotrophs which means that they obtain energy from inorganic substances. If there is life on another planet or moon, it is likely bacterial.
Another good argument for alien life existing somewhere in the universe is that the universe is huge. Carl Sagan famously said that there were more stars in our universe than grains of sand on the Earth's beaches. If each star has even a few planets revolving around it, that is a staggering number of planets that exist, and it is still a staggering number of "Earthlike" planets. Even if you said that the chances are less than one percent that another planet has life on it, that is still an unfathomable number of planets within that small percentage because the numbers are astronomically huge.