What are current understandings of the heliocentric theory?
In astronomy, heliocentrism is the theory that the Sun is stationary and at the center of the universe. The word came from the Greek (ήλιος Helios = sun and κέντρον kentron = center). Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center. Discussions on the possibility of heliocentrism date to classical antiquity. It was not until the 16th century that a fully predictive mathematical model of a heliocentric system was presented, by mathematician and astronomer Nicolaus Copernicus. In the following century, this model was elaborated and expanded by Johannes Kepler and supporting observations made using a telescope were presented by Galileo Galilei.
The current understanding is that the sun is an ordinary star fairly far out (about 2/3) in a spiral galaxy (Milky Way) that is composed of about a billion billion stars. The Milky Way is an ordinary galaxy out of about a billion billion galaxies in the observable universe. So the sun is only the center of our planet system, not the center of any thing else in particular. In fact this is a well respected principle, called the Copernican principle, which states that our position in the universe is typical, in no way special on cosmological scales (except possibly for intelligent life at this time that can ask such questions).