Geocentrism claims the sun, the planets and all the stars revolve round the Earth while the earth stays fixed at the centre of the Universe.
Geocentrism takes different forms. Literally "Geocentrism" simply means the earth is at the centre and that includes people who believe the earth rotates daily without leaving the centre of things. Geocentrists must believe the earth stays at the centre of the universe and therefore doesn't circling the sun yearly.
The traditionally geocentrism was that the earth doesn't remains completely fixed, that the whole Universe rotates round the earth daily. So traditional geocentrists imagine stars and galaxies many light years away rushing round the earth daily many times faster than light. How silly can they get? People can also take a relativistic view, that "all motion is relative", and claim we can't tell the difference between the earth being in motion and it being fixed. This has the fault that, instead of the Bible making false statements about the the earth being fixed, the Bible is making meaningless statements. Therefore believers rarely say this.
The earth seems fixed and unmoving and in earlier times people didn't know the earth moves.
1 Chronicles 16:30: "He has fixed the earth firm, immovable." 
Psalm 93:1: "Thou hast fixed the earth immovable and firm ..." 
Psalm 96:10: "He has fixed the earth firm, immovable ..." 
1 Chronicles 16:30: Fear before him, all the earth: the world also shall be stable, that it be not moved.
Biblical writers didn't understand astronomy but the God who inspired it was supposedly Omniscient. So why are there glaring errors? Further the psalmist is telling us that we have as much trust in the Lord as we have in the The earth being fixed. If the earth is not fixed, then what reliance can we have in the Lord?
It's difficult to argue with a convinced person about how to interpret the Bible, what is meant literally and what is figurative. Still we know that nobody before the time of modern science thought that the Bible could mean the Earth is a planet of the Sun. It's clear that no one has ever arrived at Heliocentrism from understanding and interpreting the Bible intelligently. It is only from scientific observations and human reasoning or by the accepting what people today generally believe that anyone even today accepts that the Earth moves. If people discuss how to interpret the Bible, believers only accept heliocentric interpretations because modern science shows them they should.
Occam's razor tells us simple explanations with few assumptions are more likely to be true. There is a simple explanation for the Bible quotes in this article, the people who wrote the Old Testament believed the earth is fixed while the sun moves. 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 Old Testament states clearly the earth is fixed immovable on firm foundations. Believers have tried to reconcile the immovable earth in the Bible with the moving earth that modern science shows. The earth appears fixed in relation to us and people who understand heliocentrism can claim the texts is figurative. Still these complex explanations are less plausible.  
Ancient Greeks and later
Throughout the Western world and later the Arab world till the end of the Middle Ages and the Renaissance educated people accepted the Geocentric theory. The Bible convinced believers it must be true. If the Old Testament was divinely inspired one may ask why an inspiring deity chose ambiguous and misleading language. Anyway Ptolemy's model of the solar system, placed all celestial bodies orbiting around the Earth, with very complex epicycles introduced to compensate for the way planets like Mars appear to move backwards sometimes.
The ancient Greeks pointed out that, if the Earth did move around the sun, the stars should appear to change position from one part of the year to the next, which the stars didn't appear to do. Either that, or the stars were enormously far away so their shift in position was tiny. Hey, that would make for a ludicrously large empty space between the most distant planet, Saturn, and the "fixed stars". As we know today stars other than the sun are Light years away and up to the 19th Century astronomers couldn't measure any change in position (parallax). Stellar parallaxes are real and since the 19th Century astronomers have detected them.
Mainstream Scientists assumed the Geocentric theory of Ptolemy was correct until Nicolaus Copernicus suggested the idea of a heliocentric universe in 1514. Incidentally his book was still listed on the Index (the list of books banned by the Catholic Church) between 1616 and 1758. This could have been because of the Galileo controversy,
Ignoramuses True believers who accept a fixed earth in the 21st century sometimes agree with the Ptolemaic system and its complications. Others prefer a different model and may call their model geocentricity trying to make that clear. They may prefer a "compromise" position first suggested by Tycho Brahe: the Sun revolves around the earth, but the planets revolve around the Sun. The Bible doesn't mention the way the planets (and the stars move), so true believers can choose any way the heavenly bodies move they like, just as long as the earth is fixed and the sun is in a closed path around the earth.
The demographics of geocentrism
Modern geocentrism is what a minority of dense fundamentalists believe. They aren't as few as you might expect and sadly include 25% of Americans.  Modern geocentrists are sometimes rather vocal and have websites,  and would like you to know that at least they're not as dumb as those heretic Flat Earthers.
The sun travels around the earth
Ecclesiastes 1:5 says "The sun rises and the sun sets, and hurries back to where it rises." Geocentrists think this means that the sun revolves around the earth. This was interpreted literally before Copernicus and used against Galileo at his trial. There is also Psalm 19:6 "His [the sun's] going forth is from the end of the heaven, and his circuit unto the ends of it: and there is nothing hid from the heat thereof.".
The point of the biblical passage that contains the verse is that life on earth amounts only to endless, monotonous, and meaningless cycles-- such as the endless cycle of sunrise and sunset. The true spirit of the passage is contained in verse 9 of the same chapter, which reads "What has been will be again, what has been done will be done again." Those who put Galileo on trial thought there were astronomical implications as well, modern Christians tend to think differently.
Ceasing the revolution of the sun
Joshua 10:12-13 reads "On the day the LORD gave the Amorites over to Israel, Joshua said to the LORD in the presence of Israel: "O sun, stand still over Gibeon, O moon, over the Valley of Aijalon." So the sun stood still, and the moon stopped, till the nation avenged itself on its enemies, as it is written in the Book of Jashar. The sun stopped in the middle of the sky and delayed going down about a full day." What would be mechanical difficulties if the sun and moon suddenly stood still? Incidentally this would only work if the sun were traveling around the earth. If the sun appeared above one city for any prolonged period the earth would have to stop rotating, it would not matter if the sun were moving or not. (See this video for the terrible things that would happen if the Earth suddenly stopped rotating.)
The dance of the stars
There's yet another reason why modern geocentrism goes against science. The rotation of the earth isn't completely smooth, there is a slight wobble. The earth's rotation speeds up and slows down slightly and the point of the earth's axis changes slightly. 
- Heliocentrists have no difficulty explaining this wobble, the earth moves.
- Geocentrists have to imagine that the stars and galaxies all wobble in synchrony 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,
- A star 100 Light years away wobbled 100 years ago,
- A galaxy 6 billion Light years away wobbled 6 billion years ago. Very distant galaxies must have wobbled before the Solar system formed.
- The light from all these wobbles that actually happened at different times reaches us simultaneously giving 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 Fall of the Geocentric Theory, and the Rise of Heliocentrism
- Man in the Middle: An Exclusive Cut Excerpt from Rapture Ready! by Daniel Radosh
- Geocentric theory
Adapted from RationalWiki
- 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. The Cassini probe also used gravity assist to reach the planet Jupiter. newton showed 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.
- Geocentrism lives Copernicanism was the first time science proved religious dogma wrong, therefore some people believe what’s wrong with the world started then.
- One in four Americans 'don't know the Earth orbits the Sun' and only half believe in evolution
- The non-moving Earth & anti-evolution web page of The Fair Education Foundation, Inc. Static Earth, Ready to have your world view shaken
- An atheist dissects and interprets the Word of God.
- What is the Chandler wobble?
- Modern geocentrism