Who created the mechanical model of the Solar System in the 100s?
I'm assuming you're talking about the Antikythera mechanism, recovered in 1900 from an ancient shipwreck discovered near to the Greek island of Antikythera
What is the Antikythera mechanism?
The Antikythera mechanism is a mechanical computer thought to have originally been built with the purpose of calculating the positions of the Sun, Moon, Mars, Venus and possibly even the other three planets known at the time of its construction, with respect to the celestial sphere and from a geocentric perspective.
Take a look at the video below (not mine) to see a modern representation of how the original article may have worked:
The Mysterious Gloved Hand is turning a handle that makes the mechanism work. On the same disc that contains this handle is a representation of the Moon and its phases. The other dials and pointers indicate where to look for certain objects in the sky, with the handle being used to set the machine to a particular date. Watch the pointers on the right-hand part of the machine carefully, and you'll notice that they occasionally stop and go backwards for a short while before returning to their original direction. This is a representation of what we actually see the planets doing from our perspective here on Earth, and is caused by the fact that we are also in orbit around the Sun as opposed to being at the centre of things with everything else orbiting around us.
Why is it so interesting?
Various sources, including carbon dating of pieces of the shipwreck and analysis of the artifact itself including inscriptions on the dials, have placed the construction of the piece at somewhere between 150 and 100 years BCE. It functions by clockwork, with the recovered piece containing over 30 gears, with some experts suggesting that the complete working machine originally containing as many as 72 different parts. This complexity along with its relatively small size puts the accuracy, understanding and workmanship of its creators in the same league as 19th century Swiss clock makers 1900 years before their time.
Without wanting to bore you to death, the Antikythera mechanism is an astoundingly complex and accurate piece of work for its time, incorporating mechanical features that, for example and among other things, are designed to compensate for the differences in the then-used Egyptian calendar and the Solar year: we cope with this by incorporating leap years into our calendar, but this wasn't introduced until around a century after this machine was built.
Who built it?
Due to factors such as the workmanship, technical skills and know-how and the Koine Greek inscriptions, it is generally assumed to have been built in the Greek-speaking parts of the world at the time. This is supported by the implications in the design and possible function of the machine that it was built around mathematical and astronomical theories known to have been developed by the ancient Greeks.
Beyond this, its origins are unclear, though there are a number of hypotheses. One is that the mechanism and its concept originated with the Corinthians, making a link with Archimedes a tempting possibility. Another, more widely discussed theory is that the machine was developed on Rhodes island by astronomers of an ancient academy founded by the philosopher Posidonius. This theory, too, has possibilities of a grand name in the history of astronomy playing a part, as the motion of the Moon seems to have been modeled using theories of the contemporary astronomer Hipparchus.
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