The Geocentric theory is the obsolete idea that the sun, moon, planets and stars all revolve round the Earth. Geocentrism imagines the Earth isn't a planet and the sun isn't a star. Geocentrism is connected with the Ancient Greek philosopher Ptolemy and was widely accepted in the Ancient world up to the end of the Middle Ages.
People who understand Science gave up the geocentric theory after the discoveries of Copernicus, Galileo and Kepler. This happened especially after Kepler's system gave better predictions for the movements of planets than previous systems. The first new Theory was the Heliocentric theory that the sun was the centre of the universe. (Today we know the sun is the centre of the Solar System only and isn't near the centre of the Milky Way Galaxy or in any special place in the universe.)
Problems for monotheists
As telescopes improved, astronomers discovered that we are a very long way from the centre of anything. This is a bit of a problem for monotheists. Supposedly we are somehow close, special and made in the image of an all seeing, all powerful God that created the universe. Then why are we on such a tiny speck of a planet, orbiting a fairly normal star lost in a vast galaxy of other stars? And that Milky Way Galaxy itself is nothing special, too, just a snowflake in a snowstorm of other galaxies. 
Liberal theologians may think along lines like, What other images and children does God have? and How splendid and diverse may be his creations. The narrow minded just think in ways like, Ah thought we was special and Is He cheating on us with E.T.?.
Atheists and, to be fair, some Polytheists, have no problem with what Astronomy implies. But a few Bible thumpers try to deny Science and insist on discredited ideas of a Geocentric Universe only because reality has become painful for them.
Modern geocentrism or egocentrism is what a minority of generally rather dense and sometimes rather vocal Fundamentalists follow.  There are fairly few of them but they include 25% of Americans. 
Ultimately, longing to have a fixed earth reflects the fundamentalist longing to recover that feeling from childhood when you just knew you were the center of the universe, and that parents, life, angels, God and all that exists revolve around you. 
The dance of the stars
There's yet another reason why modern geocentrism goes against Science. There is a slight wobble in the earth's rotation, it's called Chandler wobble. The earth speeds up and slows down its rotation a little and the point of the earth's axis changes just a bit too. See What is the Chandler wobble?. It's a good Science Web site explaining Chandler's wobble.
- Heliocentrists have no difficulty explaining this wobble, the earth moves.
- Geocentrists have to imagine that the stars and galaxies all wobble together while the earth stays still. Each wobble started at a different time so the light from these different objects reaches us at the same time giving the appearance of synchrony. 
- A star 9 Light years away wobbled 9 years ago,
- Stars 600 Light years away wobbled 600 years ago,
- Galaxies 6 billion Light years away wobbled 6 billion years ago. Galaxies that are very far away wobbled before the Solar system formed.
- The light from all these wobbles that really happened at different times reaches us at exactly the same time so there's the illusion that the whole universe wobbles in synchrony.
Is all the wobbling in the whole universe centred round earth plausible? Consider Occam's razor. Isn't the simpler assumption that the earth wobbles?
The earth is fixed and unmovable
As we know, the earth doesn't stay fixed in one place but instead rotates daily and orbits (or revolves) every year around the sun. The Bible however, goes against modern knowledge, and makes it clear in many passages that the earth is fixed and cannot be moved:
- "He set the earth on its foundations; it can never be moved."(Psalm 104:5)
- "The world is firmly established; it cannot be moved."(Psalm 93:1)
- "Say among the nations, ‘The Lord reigns.’ The world shall be firmly established; it cannot be moved."(Psalm 96:10)
- "Tremble before him, all the earth! The world shall be firmly established so it cannot be moved."(1 Chronicles 16:30)
Occam's razor favors the simplest answer. The Old Testament authors simply believed the earth is fixed and wrote what they believed. The simple explanation is plausible because people living in that region at that time didn't know the earth moves.
The Bible and science
- The Bible doesn't say the earth orbits and rotates in ways impossible for humans to change.
- The Bible says the earth has firm foundations and isn't moving.
Believers try to reconcile the immovable earth in the Bible with the moving earth that modern Science shows is real. This involves complex unprovable assumptions. Remember Occam's razor tells us we should prefer simple explanations with few assumptions rather than complex explanations with many assumptions.
- Believers must assume the Bible is divinely inspired while other supposed sacred texts from other religions that contradict the Bible are false Superstition. There is no reason to treat the Bible differently from other supposed sacred texts.
- Believers must assume the inspiring Deity chose ambiguous language open to misinterpretation. Especially before Heliocentrism came to be widely known the Bible seemed to show a fixed earth. Why should a deity go to the trouble of inspiring a sacred text and leave the language unclear?
- As shown above the Bronze Age and Iron Age people who wrote the Old Testament didn't know the earth moves and the simplest explanation is that they wrote what they mistakenly believed.
- That makes sense since people living in in Judea then didn't know anything nearer the truth. The earliest known heliocentrist was Aristarchus and he lived in a different area after the sections of the Bible quoted were written. 
- The simplest explanation is that the Bible is mythology similar to other Mythology of the time and that the Bible is untrue. 
- Geocentrism imagines the earth is somehow special and different at the centre of the universe. Earth may be specially bad and Hellish, after all Hell is even nearer the centre of things. Fundies spend part of their time imagining they're superior made in the image of God, other times they imagine the earth is a low place and we need God's mercy to ascent higher to Heaven. Either way they find it hard to see themselves as ordinary on an ordinary planet orbiting an ordinary star.
- Geocentrism, Young Earth Creationism and Literalism
- One in four Americans 'don't know the Earth orbits the Sun' and only half believe in evolution
- Fixed Earth: Setting the Record Straight
- Modern geocentrism
- Does the Bible teach that the earth is 'fixed' and 'immovable'?
- The Defender's Guide for Life's Toughest Questions Ray Comfort wrote Bible texts saying the earth is immovable mean we cannot move it. The Earth's Gravity was used to alter the paths of spacecraft, see Gravity assist. The Galileo spacecraft used the earth's gravity twice, see Galileo – a change of plan. Every action has an equal and opposite reaction. If the earth's gravity changes the speed and direction of a spacecraft the earth's speed and direction of movement change very slightly. This is real and can be calculated, humans have literally moved the earth.
- Historians know of no earlier heliocentric theories than the theory of Ancient Greek astronomer, Aristarchus of Samos. In any case Aristarchus didn't have much of a following and heliocentrism didn't become popular till Copernicus, Galileo, and Kepler made their discoveries millennia later.
- One thing that's really hard to explain is why in the 21st century so many people still take seriously texts written by Bronze Age and Iron Age people who didn't even understand the dynamics of the Solar system.