There are many theories people have come up with to try to explain how our solar system came to exist. The most accepted in the scientific community is the Nebular Hypothesis.
The Beginning
With the Nebular Hypothesis, the solar system started out as a ball of gas [mostly not the smelly kind, thankfully]. This ball of gas will become disrupted by a strong force that has occurred nearby (i.e.: a super nova). This will cause the ball of gas to start collapsing because of its own gravity. From it pulling in on itself with its gravity, it will create a core; this core will be denser and hotter than the rest of the cloud of dust and gas. The initial collapse of the nebula will take less than 100,000 years to complete.
At this point in development, the center of the cloud has become a protostar. The protostar will have all of the gas, dust, and other debris orbiting around it. The debris around the protostar doesn't just sit there, though [I hope not. I hate couch potato dust], it also rotates around the protostar. At the same time it is rotating around the soon-to-be star, it is continually being pulled in by its gravity. The two of these forces combined cause a centrifugal force (meaning that it is directed away from the protostar). This ends up causing an accretion disc. The accretion disc will eventually spin and cool away the majority of its energy.
At this point the gas that is orbiting the protostar could collapse under the gravity of itself and thus create a double-star (this is actually more common in solar systems than the single star like we have in our solar system). If this does not occur than the gas will cool off enough to allow metal, rock, and other debris to condense out into small particles (the metal will condense sooner than the rock by approximately one billion years).
Soon after the particles have cooled, they will begin to collide with each other and begin to form larger objects. The larger the objects get, the more gravitational pull they possess, the larger it gets in size, and at a faster rate in which it increases in size. These objects can get to huge sizes; as big as moons or planets. And as the further away from the protostar that these objects are when they form, the bigger they get.
The Final Stage
After another million years [long line to become a star] the nebula has cooled and the star will generate a [not small, medium, or big, but] huge solar wind that will wipe out all of the gases off of all of the spatial bodies (the soon-to-be planets and their moons). When all the gases get sweeped away, its up to the spatial bodies to try to get them back. In the end, the ones that were able to pull back all or most of their gases (or an equivalent amount from bodies that couldn't pull theirs back) will become the gas giants, the ones that are unable to bring back their gases will become rocky or icy planets.
Now, you have a fully developed solar system with a star, approximately ten planets, their moons and other bodies (the asteroids, comets, and meteorites).
