This is a good question. This topic is hugely important in science and the philosophy of science, and it was the basis of a scientific revolution. I will give some background on the topic before summarizing some observations that strongly support the theory that the earth orbits the sun. Most of the information that I am presenting below is from the NASA website

There has been debate since the time of the ancient Greeks whether the earth orbits the sun or the other way around. Aristotle, a prominent Greek philosopher, hypothesized that the earth is the center of the universe, and that all other celestial bodies rotate around it. This was based on observations such as:

1) an observer cannot feel the earth move,
2) there was no perpetual wind on the surface of the earth. The geocentric (earth in the center) view of the universe prevailed for over a millennium.

In the 1500s a Polish scholar, Nicolaus Copernicus, proposed a heliocentric model: that the sun was the center of the universe. Keep in mind that we now know that the sun is not actually the center of the universe. For a while, the church oppressed Copernicus’ hypothesis because it contradicted certain dogmatic beliefs.

After Copernicus died, his hypothesis began to gather support, and strong scientific work by Johannes Kepler and Galileo provided evidence that finally dispelled the geocentric model (at least among scientists). Kepler formulated a mathematical model for how earth and the other planets of our solar system orbit the sun.

Galileo used the first high-powered telescope to observe planetary motion. One of his most important observations was that there were several small bodies that orbit the planet Jupiter. This was a very important observation that ruled out the idea that all bodies orbit earth. Another one of Galileo’s observations was that Venus displayed a full range of phases (like the shadows on our moon), which can only be explained if Venus and earth both revolve around the sun on different orbits.

Isaac Newton’s theory of universal gravitation tied everything together. It provided a strong theoretical basis for why the earth and other planets orbit around the sun.

So what about Aristotle’s observations? They are accurate, right? We can’t feel the earth hurtling through space. Here’s a thought exercise:

The earth revolves about its axis about once every 24 hours. This is the angular velocity of the earth about its rotational axis. The radius of the earth at the equator is about 3960 miles. The magnitude of linear velocity of a rotating point is equal to the angular velocity (in radians/time) multiplied by the radius to the point. Calculating the velocity of the surface of the earth at the equator relative to the center of the earth:

(2π radians/24 hours) x 3960 miles = ~ 1037 miles/hour

That’s pretty fast. Why don’t we feel it? It is because speed is always measured relative to some reference frame. The surface of the earth is the reference frame relative to which we sense “speed”.

The bottom line here is that the model of the earth and other planets revolving around the sun fits our observations. The model that the earth is the center of the universe is inconsistent with our observations– they do not fit. A crucial characteristic of science is that hypotheses (like the geocentric model) must be refutable.

That is, there must be ways to test if they are wrong. If a hypothesis is proven wrong by our observations, we must be prepared to change it or discard it. Keep in mind that this is how science works, and religious beliefs do not evolve in the same way as scientific beliefs. Religious and scientific beliefs are not really comparable. When religious beliefs are considered in a scientific way, they are easily refuted. This doesn’t mean that religion is bad, it is just fundamentally different from science.