Here's my answer. I'll try to make it as comprehensive as possible.
It's pretty hard to define the edge of the Solar System. Most people would probably define it as where objects are no longer gravitationally bound to the Sun. That just shifts the question a little, though: Where is that dividing line? To try to answer this, I'll go over the regions of the Solar System.
The first region is the domain of the inner planets - basically everything from the asteroid belt inwards. It is comprised of Mars, Earth, Venus, Mercury, their moons, and all the smaller objects that surround them. The inner Solar System is very rocky, as one can imagine. The terrestrial planets are primarily made of rock, as are the asteroids and the inner planets' moons.
The second region is the domain of the gas giants. It consists of Jupiter, Saturn, Uranus, Neptune, their moons, ring systems, and assorted smaller bodies, such as Trojan asteroids. The gas giants had a big influence on the Solar System when it was first formed, pulling in chunks of rocks, grabbing moons, and possibly stabilizing or de-stabilizing orbits. Some may have migrated outwards (as per the Nice model), but their orbits are currently stable. The gas giants are made largely of gases, but it is thought they have solid or molten cores. The composition of their moons is familiar - more like objects in the inner Solar System.
Next up is the Kuiper Belt. It's sometimes introduced as a cousin of the asteroid belt, but that's not accurate. The bodies that make up the Kuiper Belt are chunks of rock and ice. Notable examples of Kuiper Belt bodies and/or trans-Neptunian objects are the dwarf planets Pluto, Sedna, Makemake and Haumea. There are also lots of smaller objects, including some short-period comets (although these are more properly part of the lesser-known "scattered disk"). While there have been theories for years about another planet out there, it is not considered likely. The Belt extends from 30 to 50 AU.
Further out still is the Oort Cloud, named after Jan Oort. Observations of objects in the Oort Cloud are extremely difficult, if not impossible, so its existence has not yet been verified. It is populated by long-period comets and smaller objects. These are also composed of rock and ice. The Oort Cloud is thought to extend up to an incredible 50,000 AU. While the other regions so far mentioned are roughly in planes, the Oort Cloud is spherical.
Some consider the far edge Oort Cloud to be the edge of the Solar System, because the majority of the mass of the Solar System is within it, but the boundary between the Solar System and interstellar space is actually thought to be within its inner reaches: the heliopause. This is generally accepted as the Solar System's boundary because it is where the solar wind meets the interstellar medium. This is often placed at 121 AU - which is where Voyager 1 passed through in 2013. The heliopause is the far boundary of the heliosphere, beyond which the interstellar medium takes control. Inside "layers" are bounded by the termination shock and the heliosheath.
In summary, while the Solar System is made of many regions, the heliopause is considered to be its outer boundary.
Once again, I welcome any and all input regarding this question and answer.