The Periodic Table
When Dmitrii Mendeleev was writing a chemistry textbook in 1869, he pondered how best to introduce the elements. He suddenly realized that some elements had similar properties, and that those similar properties popped up at certain intervals of atomic weight. Those intervals, or periods, became the basis for classifying all elements, and no discussion of chemical properties can be complete without discussing the periodic table. For example, the four types of elements are neighbors in the periodic table.
Electronic Properties
The physical principles that determine the way electrons are structured are far too complex to detail in a short article, but the general rule is that electrons are limited to being in certain "states," and they will put themselves in the lowest energy state possible. The energy states come in "shells," and it almost always takes less energy to fill a shell than it does to jump to the next shell. So each horizontal row of the periodic table corresponds to a shell filling up. Atoms on the right edge of the table have filled up their outermost shell, while elements on the left edge have just put the first electron in a new shell. That's what drives the chemical properties.
Metals
The first type of elements are those with electrons that are loosely held in their outermost shell. These mostly have one or two electrons in their outermost shell, but a handful of these elements have three, four or five electrons in their outer shell. These outermost electrons can leave their atoms and roam around conducting heat and electricity: those are main group metals. As the number of electrons goes up, the complexity of the shells goes up, too; so the whole middle section of the periodic table has loosely held outer electrons while their inner electrons build into more complex shapes. Those are also metals, called transition metals. Carbon is not a metal.
Metalloids
As you drift towards the right on the periodic table, on the edge of the group of metals you will find a type of elements that sometimes act like metals, and sometimes don't. Elements like silicon and germanium are in this group. This type is called metalloids. It's a nebulous definition, that is really all encapsulated by the phrase that metalloids are kind of like metals, and kind of not. Carbon does not fall into this type.
Nonmetals
The upper-right triangle of the periodic table consists of the type called nonmetals. Metals are pretty well defined. They conduct electricity and heat, they're kind of shiny, they can be molded and bent. Nonmetals are very ill-defined. Some are crumbly, some are hard, some are gases, some are solids. Their properties --- dictated by their electronic structure --- are common only in so far as they are all different from metals. Carbon is a nonmetal.