Great question. Precipitation is any form of water that falls from the sky. This can include rain, snow, hail or sleet. Usually we measure liquid in terms of volume, like one gallon, or one liter. Volume is really just 3 units of length multiplied by each other. For example, 1 liter is 1000 cm^3, or if you would like to think of a cube of water, it would be 10 cm by 10 cm by 10 cm. The volume is 10*10*10 = 1000 cm^3. However, when we talk about precipitation, we do it in terms of length. It's a little strange, but it's because we talk about how much precipitation falls over some area, the area that got rained on. Area is length*length. So if we want volume per area

(length*length*length) / (length*length)
we end up with just a single length. That means that we measure rainfall in terms of inches. You can think about it as the thickness of water that would be left on the ground if none of the rain washed away or was absorbed by the ground.

We can measure this in probably the way you would think. Set out a flat container, and measure the height of the rain that collects in it. That's as easy as it is. Of course there are fancier tools that this. For example, you can use a funnel that collects a lot of rain, and then funnels it into a thin tube. If the area of the funnel opening is 10 times larger than the tube then you will get a height that is 10 time more than the rain that actually fell. In this way, scientists can measure small amounts of rain very accurately.

Snow and hail and other types of precipitation can be measured in much the same way. You can collect the snow in a flat bottomed contained and measure how high it is. however, snow (and other forms of precipitation) can be light and fluffy or they can be wet and heavy. And so another way to measure these forms of precipitation is to melt them and measure what the equivalent height would have been if it had been just rain that had fallen.

Precipitation is measured in units of depth, e.g. centimeters, inches, etc. So, for example, a layer of snow three centimeters deep is measured as three centimeters worth of snow. Precipitation from multiple storms does add up, and measurement vessels are often buckets so that precipitation doesn't run off and make it otherwise hard to measure (as rain does).

Thank you for the question.
You can measure precipitation by using something like a rain gauge also known as odometer, pluviometer, or ombrometer. It is mainly measured in millimeters (although sometimes inches or centimeters if you were curious).