Thanks for your fascinating question! Hundreds of books and journal articles have been written on the subject of dating rocks. How to boil this down to a manageable answer?
There are two fundamentally different ways of dating rocks, relative methods, and "absolute" methods.
In relative dating, scientists work out the sequence by which various bodies of rock came into existence. We try to figure out what unit came first, what next, what after that one, and so forth, establishing their relative age from oldest to youngest. There are many ways of determining such sequences. For example, in layered rocks, such as at the Grand Canyon, those at the bottom of the stack are older than those at the top.
Another way we can determine the relative age of rocks is through the fossils they contain. For example, we have determined that rocks containing the remains of T. Rex are older than those that contain fossil elephants. Many other tools/observations allow us to establish the relative ages of rocks as well, but I hope this gives you an idea of the general process. Relative dating was used to construct the famous geological time scale. For example, Paleozoic, Mesozoic, and Cenozoic refer to the fossils (ancient life:"zoic") found during those time intervals.
With absolute dating, scientists measure the amount of certain chemicals in certain kinds of rocks to determine age in actual years. Even in Darwin's time, we knew that the Mesozoic (middle life) was younger than the Paleozoic (ancient life), but we didn't know in millions of years, what time intervals they spanned. An analogy would be if you have a sibling. It's useful to know who is older or younger, but it's equally useful to know when the two of you were born, and how many years/months/days separate the two of you. Both ways of "telling time" are useful. I could tell someone that I have a younger sister (relative dating), or I can say, my sister was born 13 months, 3 days, 17 hours, 48 minutes, 6 seconds after me (absolute dating), depending on what somebody was trying to determine.
So...how do we date rocks absolutely? Imagine that a certain kind of potassium occurs in some crystals within rocks. Those crystals grew as the rock they're in formed. This kind of potassium radioactively decays into a certain kind of argon the longer that crystal sits around. In a laboratory we can measure how fast this radioactive decay occurs. Therefore, by measuring how much of this special potassium and argon there is in a crystal, and knowing how fast the transformation of one into the other occurs, we can calculate how old that crystal (and rock) is in millions of years.
There are many other radioactive substances (isotopes) in nature, that are used for absolute dating. The most famous of these is Carbon 14, which is commonly used to date archeological artifacts.
There you have it...the basics of how geologists date rocks, a subject perhaps more complicated that you originally imagined.
Stay curious about rocks, and be well.
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