I'll tell you the very basics of what a carry is.
A carry is a champion that usually has awesome late game potential/ is awesome when fed or farms lots of gold.
It is called a carry because usually, if fed enough, he can literally "carry" the team to victory even if they are deep in the sh**.
With this definition, people often debate what heroes are actually carries. But, it's generally clear I think. For instance, Pantheon (even pre-nerf) is not a carry. He is a melee champion, but he generally is most useful early and mid game, and then putters out late game, no matter how fed.
Examples of actual carries (I'll give a wide range) are Tryndamere, Ashe, Gangplank, and Ryze. As you can see, ranged or melee heroes can be 'carries' depending on how well their abilities work out late game when fed. Usually they can dish out punishing damage.
What is the most important role in solo queue? Well, it's generally the carry.
All roles are important, but as a carry, like I said, you can pull your team out of the muck. A tank is essential to have, especially a skilled tank. But you put an expert tank on a team that is full of fail, and the tide will not be turned too much.