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Geocentrism is the belief that the sun, the planets and all the stars revolve around the Earth. You can thank the Old Testament for the lingering belief in this one. It has at least four verses claiming that the earth "cannot be moved," and one stating that the sun goes around the earth.
There are possible variations on this. "Geocentrism", taken literally, means that the earth is at the center, which is compatible with the daily rotation of the earth, with no annual motion around the sun. However, this option is not the traditional understanding, nor the belief as it persists today, which is that the earth has no motion at all, which means that the entire universe revolves around the earth daily. (This would mean that the stars and galaxies revolve around the Earth once every 24 hours, despite modern astronomy's assertion that they are many light-years away and would thus have to be whizzing around faster than light.) It is also possible that a person could take a relativistic view, that "all motion is relative", claiming that one cannot tell the difference between the earth being in motion and it being fixed. This option would have the fault that, instead of the Bible making false statements about the fixity of the earth, it is making meaningless statements; and therefore it is seldom taken.
This theory was accepted as standard throughout the Western world through the Middle Ages and the Renaissance. It was held up by Scripture and Ptolemy's model of the solar system, which placed all celestial bodies orbiting around the Earth, with epicycles introduced to compensate for the retrograde motion of planets such as Mars. The ancient Greeks noted that, if the Earth did move around the sun, the stars should appear to shift their positions from one part of the year to the next, which the stars don't do. Either that, or the stars would have to be so ridiculously far away that their shift in position was too small to be seen by the naked eye, and hey, that would make for a ludicrously large empty space between the most distant planet, Saturn, and the "fixed stars". (Measurements dependent on naked-eye instruments could detect a parallax of something like one arc-minute, which would give a distance roughly 300 times the distance to Saturn.)
The theory was not formally challenged until the works of Nicolaus Copernicus, who suggested the idea of a heliocentric universe in 1514. However, probably as a consequence of the Galileo controversy, his book was still listed on the Index (the list of books banned by the Catholic Church) between 1616 and 1758.
21st century believers in a fixed earth may distance themselves from the Ptolemaic system of epicycles. Some call their model geocentricity in an attempt 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. There is no interest shown by the Bible in the motions of the planets (and the stars), so one has freedom in choosing any description of he movements of the heavenly bodies, as long as the earth is fixed and the sun is in a closed path around the earth.
 For the Bible tells us so
In order of appearance, the verses are:
- 1 Chronicles 16:30: Fear before him, all the earth: the world also shall be stable, that it be not moved.
- Psalm 93:1: The LORD reigneth, he is clothed with majesty; the LORD is clothed with strength, wherewith he hath girded himself: the world also is stablished (sic), that it cannot be moved.
- Psalm 96:10: Say among the heathen that the LORD reigneth: the world also shall be established that it shall not be moved: he shall judge the people righteously.
- Psalm 104:5: (Bless the LORD . . .) Who laid the foundations of the earth, that it should not be removed for ever.
- Ecclesiastes 1:5: The sun also ariseth, and the sun goeth down, and hasteth to his place where he arose.
Be aware that the psalms were songs sung in praise of Jehovah. Psalms 93 seems to thank Jehovah for the floods (of Noah?). Psalms 96 commands the listener to pray to Jehovah because the gods of other nations are false (idols). Psalms 104 thanks Jehovah for all he created and for his protection against outsiders.
However—and this is a point that bible fundamentalists overlook—nowhere does the psalmist or any other biblical writer claim to be an astrophysicist. On the other hand, they point out that the psalmist is telling us that we have as much trust in the Lord as we have in the fixity of the earth. If the earth is not fixed, then what reliance can we have in the Lord?
Yet it is difficult to argue with a convinced person over the proper interpretation of the Bible, what is to be taken literally and what is figurative. What we can say is that nobody before the rise of modern science thought that the Bible was compatible with the Earth being a planet of the Sun. A couple of late medievals (Nicholas of Cusa and Nicolas Oresme) brought up the possibility of the motion of the Earth. It is clear that no one has ever arrived at heliocentrism from enlightened Scriptural understanding. It is only from scientific observations and human reasoning or by the acceptance of common opinion that anyone even today accepts the motion of the Earth, and if the interpretation of the Bible is brought up, the heliocentric interpretation is a result of the acceptance of modern science on that issue.
 The demographics of geocentrism
Modern geocentrism is the domain of a minority of particularly dense fundamentalists. They are few in number but rather vocal, 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." This is understood by geocentrists to mean that the sun revolves around the earth. However, the most common and contextually-informed understanding of the verse is that it is describing the visible path of the sun in the sky, not the path of the sun in the universe. 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." It is clear from this perspective that the verse makes no astrophysical implications whatsoever.
A similar verse is Psalm 19:6 which reads "It [the sun] rises at one end of the heavens and makes its circuit to the other; nothing is hidden from its heat.".
 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."
Aside for the purely mechanical difficulties of having the sun and moon stand still this would only work if the sun were traveling around the earth. For the sun to appear above one city for an extended amount of time 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.)
Galileo, in his Letter to Castelli pointed out that in the Ptolemaic system of the universe, the heavens in general are moving around the earth from east to west, while the motion of the sun in the heavens was a motion from west to east; and therefore, if the Bible were describing a Ptolemaic universe, and saying literally that the sun ceased its motion, the result would be less daylight (by about four minutes).