This is one of the most important questions in geology!

To understand the answer to this question you need to know about radioactive elements. These are elements that slowly break down to other elements over time.

If fossils are younger than 50,000 years old we can measure their age using radiometric dating of carbon. Some carbon breaks down over time and so if we know how much carbon has broken down, we can calculate how long it has been since that animal died.

But after 50,000 years nearly all of the radioactive carbon will be gone, because it breaks down too fast.

Instead we can measure the age of some minerals, especially minerals from volcanic rocks. When a volcano erupts it can create lava flows (like in Hawaii) or send out big clouds of ash that can rain down hundreds of miles away (like Mt St Helens). Those volcanic rocks have minerals that we can measure the age of, because they have elements that take longer to breakdown than carbon.

For example if we know the age of volcanic rocks below the fossil and above the fossil, we know that the age of the age of the fossil has to be something in between.

There are two ways. One is the decay of radioactive elements. Radioactive elements like uranium, which exist naturally in (very) small amounts in volcanic rocks, decay at a known rate. As a result, by measuring how much uranium there is left in a rock, we can calculate how long ago that rock was formed. Also, different kinds of animals and plants lived at different times in the Earth's history. By finding fossils of these animals and plants, we know when the rocks containing those fossils were laid down, because the life that made the fossils had to be alive at the time those rocks were formed.