This is a very good question, one that is still a topic of scientific research. Here is the basic theory:

The solar system formed from a cloud of cold gas that collapsed due to gravity (gravity is the force that keeps us attached to Earth). A big glob (semiliquid substance) of stuff formed in the center and eventually became the sun.

Meanwhile, some of the cloud material orbited around the proto-sun (the gaseous cloud that underwent gravitational collapse to form the sun) and flattened into a disk. In the disk, some matter came together to form small planetoids that slowly grew.

The matter that was closer to the center was also warmer so only the more dense stuff such as metals and rocks combined together to form planets; the warm gas was moving too fast to get caught.

Farther away, everything was cooler so gases like hydrogen and helium could also get sucked up by the new planets. So the planets closer to the sun (Mercury, Venus,Earth, and Mars) are small and rocky while the ones farther away (Jupiter, Saturn, Uranus, and Neptune) are big gas giants.

The rings seem to be partly made of frozen gases which don't exist closer to the sun. So the bottom line is that the farther away gas giants are much more likely to be able to form and keep rings than the inner rocky planets.

Neptune has rings too. In fact Saturn is not the only planet in our solar system that has rings, all the giant gas planets have them: Jupiter, Uranus and Neptune. However, these other ring systems are extremely thin and almost impossible to see. Planets like the Earth, Mars or Venus are made of rocky material and have no rings.

The formation of rings around planets is still a matter of debate. Two theories are that the fragments which comprise them are (1) remnants leftover from the formation of the planet which, due to gravity from the planet, were unable to coalesce into a moon; or (2) the pieces of a moon which was destroyed, such as by an impact. While the best-known rings of our solar system, those of Saturn, are primarily water ice, the rings of Uranus are darker. This suggests that they contain more rocky material than those of Saturn, supporting the hypothesis of formation by breakup of a moon or planet.

Another mystery is why the outer, gaseous planets of our solar system have rings, but the inner, rocky planets do not. As discussed here and on ScienceLine here, this may stem from activity early in the history of our solar system. Essentially, the solar wind from the sun pushed the lighter elements out of the inner solar system. The remaining heavy elements formed the rocky planets there. In the outer regions, lower temperatures made moon formation easier.

These moons apparently help to keep the ring material around the planets (and breaking up the moons may replenish ring material which does escape). As a result, the outer planets are believed to have formed and retained rings while the inner did not.

In general, the other planets are too small and distant to have significant influence over conditions on Earth.

However, recent evidence suggests that Venus (and Jupiter) slightly warp the orbit of Earth such that there are cycles in the climate. These two have an effect because Venus is the closest planet to Earth and is decently massive (only slightly less than Earth), while Jupiter, though more distant, is so massive that the gravitational effects are detectable The cycles affect temperatures (e.g. hotter or colder summers) as well as periods of drought/wetness.

These cycles have a period of around 405,000 years, with the current cycle probably beginning around 200,000 years ago.