What made us think that Earth moves around the Sun? Trying to observe the night sky for a few weeks, the motion of the Sun and the stars pretty much fits into the Geocentric Theory i.e. All of them move around the Earth.
What then, which particular observation, made us think that it could be the other way around, that all the planets move around the Sun?
 A: It would seem so, but there are a couple of subtleties involved.
Firstly, if you time the rise and set of the sun everyday, and the rise and set of your favourite stellar constellation, you'll find that the constellation rises four minutes earlier and earlier everyday. This is easily explained if the Earth is revolving around the sun, but not so if the other way around.
Also, if you observe the motion of comets you'd have to device some really fancy trajectories for them to be revolving around the Earth, rather than the Sun. The same goes for planets, which don't follow the same path "around the Earth" as the stars do. It just made more sense overall to allow for the Earth to be revolving around the sun.
A: The observation of the phases of Venus through early telescopes by Galileo Galilei were key to discrediting the geocentric model. Search the web for "phases of Venus in geocentric model" for many good summaries of this history. Paraphrasing what I've read: 
Galileo Galilei used his telescope to observe that Venus showed all phases, just like the Moon. He thought that while this observation was incompatible with the Ptolemaic system, it was a natural consequence of the heliocentric system.
Ptolemy placed Venus' deferent and epicycle entirely inside the sphere of the Sun between the Sun and Mercury. He could just as easily have made any other arrangement of Venus and Mercury, as long as they were always near a line running from the Earth through the Sun, such as placing the center of the Venus epicycle near the Sun.
Under the Ptolemaic system, and if the Sun is the source of all the light: If Venus is between Earth and the Sun, the phase of Venus must always be crescent or all dark. If Venus is beyond the Sun, the phase of Venus must always be gibbous or full.
But Galileo saw Venus at first small and full, and later large and crescent. 
This showed that with a Ptolemaic cosmology, the Venus epicycle could be neither completely inside nor completely outside of the orbit of the Sun. As a result, Ptolemaics abandoned the idea that the epicycle of Venus was completely inside or outside the Sun.
A: 
which particular observation, made us think that it could be the other way around

Retrograde motion must be a prime candidate.

As seen from Earth against star background, Mars occasionally slows down and goes backwards. Our moon doesn't.
It probably became clear to people constructing orreries that heliocentric models were enormously simpler and more convincing. They also tied in with simple inverse square laws of gravitation and planetary motion.

The discovery by Galileo Galilei of Jupiter's moons also provided firm evidence of the existence of heavenly objects that, perversely, did not orbit the Earth.

Photo: Thomas Bresson (Galileo probably didn't have a Nikon / mobile phone handy)

Luckily, he had available a corner of a napkin, a goose and some soot (or equivalents)
