Methane is constantly produced in the earth's biosphere, by the decomposition of organic material, sometimes assisted by the fermentation processes of various bacteria and archaea, sometimes living freely, sometimes living in symbiosis with other creatures, ranging from termites to cattle. Since organic materials are carbon chains, and methane is just the simplest form of carbon chain, it makes sense that it would be a byproduct.
But once it reaches our atmosphere, it only exists for a limited time. The quickest source I could get lists the half-life of atmospheric methane as six years. This has probably varied across the earth's history, depending on temperature and atmospheric oxygen levels. But, overall, since methane is a reducing chemical, and we have an atmosphere that is an oxidizing atmosphere, at some point, they will have to meet and neutralize and produce water and carbon dioxide.
This is basic chemistry and earth science. Where methane gets interesting is in the rapidly growing field of astrobiology. Because methane's existence involves being pushed up a thermodynamic hill, there is a limited amount of reasons it could exist inside a planet's atmosphere.
- It is constantly being created by some process. One of which would be life, but it can also be created by, for example, ultraviolet light interacting with water and carbon dioxide. However, Life is the most likely process to create methane.
- The planet is cold enough that even if methane and oxygen are both present, the reaction between them is occurring at a slow enough rate that they can both exist indefinitely. This is the case, for example, on the gas giants of our own solar system.
- The methane exists with no oxidizer to counterbalance it. It forms the atmosphere, and is thus stable.
When the Curiosity Rover went to Mars, it carried equipment to detect atmospheric methane. If methane does exist in quantity on Mars, it might be a byproduct of deeply buried microbial life, slowly releasing it into the atmosphere. Of course, since on Mars there is no oxidizing atmosphere, the only way it would be destroyed is through photodisintegration. Currently, the experiments are not turning up elevated levels of methane, but if they do, the argument about whether it is biogenic will certainly be fascinating.
Much further away, the hunt for exoplanets is now reaching the point where scientists are beginning to be able to take a spectrum of an exoplanet's atmosphere, which is an incredible feat. And if methane shows up in the spectrum of some exoplanets, it would be a sign that the planet could have life. A small, rocky planet in the habitable zone of a star with both oxygen and methane in its spectrum would either have some form of life, or else very unusual non-biological chemical processes. NASA currently has a mission, called FINESSE, to analyze atmospheric spectrums for signs of life.
And so it is that the first sign of life in the universe we might ever get is the spectrum of methane, one of the simplest and most common of molecules, barely detectable in the atmosphere of a planet orbiting a star scores of lightyears away.