One of the best ways to start to explain the rain is a quote from St. Augustine, on the subject of time: "I know exactly what it is, until I try to explain it." Rain could simply be described as water that falls from the sky, but that is both inadequate, and also untrue: pure water, by itself, can not form rain. And some of the other things that fall with that rain, while minute, have gigantic long term consequences.
The first step in the formation of rain is evaporation, which mostly happens over the oceans, where most of the water of our planet exists. Due to both convective or radiative heating, water molecules near the top of the ocean gain enough energy to bounce into the atmosphere. These molecules are water vapor, which has a technical meaning: each molecule of water is alone, not bound to other molecules of water. In common parlance, mist might be called a "vapor", but the droplets of water in mist, although small by our standards, are still gigantic by molecular standards. True water vapor is different. In "dry" air, that is, air with no liquid water in it, there can still be quite a bit of water vapor. A typical cubic meter of air, which weights about 2 kilograms, can hold 10 grams of water vapor. Actually, it can hold quite a bit more or less than that, depending on temperature and pressure, but 10 grams is a good spherical cow to use. Using the magical power of the cube, a cubic kilometer of air has 1 billion cubic meters, meaning that in a perfectly clear, dry day, there may be 10 billion grams, or 10 million kilograms, or 10,000 tons of water suspended in each cubic kilometer of air.
What then happens is that that mass of air somehow undergoes a change. Most likely, it cools off, and the water vapor can now condense into actual water. It may cool off because it is night time, because it has ran into another mass of cold hair, or because it has been lifted up by terrain or by convection. Now, that "dry" vapor turns into liquid water. One of the curiosities of the rain, mentioned above, is that rain can't form directly from water molecules sticking to each other. Instead, the water starts forming around some foreign particle that has a minute electrical charge, such as dust or bacteria. Often, rain actually first forms as snowflakes, and then melts as it falls. The rain will continue to fall until the mass of air is no longer saturated...that is, until the water vapor in it can go back to being alone, without clumping into droplets. What that amount of rain is depends on the situation, from droplets that have a terminal velocity of inches an hour and evaporate before they hit the ground, to the twenty or even thirty inches that can fall on mountain peaks during a hurricane.
The fact that rain brings water and moisture is a fact that I obviously don't need to explain, but some of the other things that rain has in it are almost as important, in the long run. Inside of rain, lies dissolved carbon dioxide, which is slightly acidic and can dissolve rocks, but also helps return carbon dioxide to the ocean. Fixed nitrogen compounds, formed by lightning or cosmic radiation, are also brought down with the rain, forming an important part of the nitrogen cycle. Rain also washes particulate matter out of the atmosphere. While the total amount of carbon dioxide, nitrogen and particulates in rain may be small, the overall effect of rain in forming a connection between the atmosphere and the ground is vital. Along with these "natural" cycles, rain can also bring down sulfuric acid, which causes acid rain. While this may be damaging in the short term, it is also another cycle that is part of maintaining equilibrium.
Thus, while rain could be seen simply as "water falling from the sky", rain is actually a complicated process of redistributing the planets thermodynamic and chemical resources.