With the Bergeron process, most precipitation starts out as snow, even though most of what we see at the ground is rain. It is formed several km high in the atmosphere, where the temperature is below freezing, and the snowflakes melt on the way down.
At least, that's what happens most of the time. Other times, if the air is cold enough, the snowflakes may not melt, and they make it all the way down to the ground as snow. And on rare occasions, other types of precipitation may result.
Here's a figure that shows the atmospheric temperature structure associated with most of the major precipitation types. You can imagine the cross section as running from north to south across a warm front.
To the north (left), the air in this example is below freezing everywhere. The precipitation forms as snow, falls as snow, and reaches the ground as snow.
A bit farther south, a small layer of the atmosphere is above freezing, but the layer of air near the ground is below freezing. As the snow falls into the warm air, it melts, but then re-freezes as it passes through the below- freezing air near the ground. It reaches the ground as frozen raindrops, otherwise known as sleet or ice pellets.
Farther south again, the layer of air below freezing is quite shallow, perhaps only a few hundred meters. The falling raindrops do not have time to freeze before they reach the ground, but when they hit the below-freezing surfaces of the Earth, they freeze on contact. This is freezing rain.
Where the air is warm all the way down to the ground, the situation is simpler. The snow melts on the way down and falls as rain. This is how most of our rain originates.
Finally, in intense thunderstorms, the strong updrafts can hold the original ice particles within the supercooled water layer for an extended period of time. As the snow gets splashed with water droplets, it forms a ball of ice which must eventually become quite heavy for it to escape the updraft and fall to Earth. This is hail.
Review question: match the following graphs of temperature as a function of height to the type of precipitation that would most likely be observed on the ground.