The earth has a orbit, but not for the sun. Similarly, all planets have orbits, but not for stars like the sun.
The stars and the planets have the gravity. For example, the sun's gravity is 274 m/s2 and the earth's gravity is 9.8 m/s^2. The gravitational force and centripetal force are main causes for the rotation of the planets. Centripetal force is provided by the gravitational force for the rotation of the planets.
Objects in orbit around the Earth with periods of 24 hours are sometimes described as 'geostationary', meaning that they hold the same position relative to the Earth as they complete a revolution with respect to the stars. But even that is a misnomer, because the pulls of the Sun and other bodies cause them to describe figure-of-eight paths with respect to the Earth below.
Lagrange described five theoretical solutions to the 3-body problem, the 'Lagrange points' at which a body orbiting one body could hold a fixed position with respect to it and to another massive body also orbiting it. These are sometimes referred to as 'stationary points', but again the term is a misnomer, because an object 'at' one of those positions will again be acted on by other objects in the Solar System, and consequently will follow a complex drift around the theoretical null position, as well as continuing to orbit the primary body. The L1, L2 and L3 points are actually conditions of unstable equilibrium, in which any perturbation will cause the orbiter to move progressively further from the theoretical position. Objects 'at' the Sun-Earth L1 point, such as the SOHO and Discovr satellites, or 'at' the corresponding L2 point, such as the James Webb Space Telescope or the Gaia satellite, are actually in motion around those points, requiring course corrections to maintain their overall position.
The L4 and L5 points, also called the Equilateral positions because they are 60 degrees before and behind the more massive orbiter, are more stable, but still acted on by more distant objects. L4 and L5 are sometimes called 'Trojan' positions because the asteroids 'in' those positions, at the Sun-Jupiter L4 and L5 points, were named after characters in Homer's Iliad. But in practise they can wander almost half-way round the Sun before returning towards those positions.
In the galactic frame, the Sun is orbiting the Galactic Centre with a speed around 200 miles per second, and an orbital period of about 230 million years. With respect to the average speeds of the nearby stars, the Sun is moving at about 12 miles per second in the direction of Vega, the brightest star in the constellation Vega. The entire Milky Way galaxy is moving towards the M31 spiral galaxy in Andromeda, and will collide with it in approximately 3.5 billion years. The entire Local Group of galaxies was thought to be in orbit around the much larger group of galaxies in the constellation Coma, but is now known to be sweeping round it as part of a much larger overall grouping called Laniakea, and even that is in motion, being pulled by an even larger more distant mass called the Great Attractor, whose nature and position have still to be fully determined.
So as a previous reply rightly says, nothing is stationary in space - everything is moving under the attraction of multiple other bodies.
No star in the Milky Way is stationary with respect to the galaxy centre. They all move along orbits around it.
Yes, the Sun does move in space. The Sun and the entire Solar System revolve around the center of our own Galaxy. https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question18.html
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