One of the main chemical properties of carbon is that it can easily bond with other elements in long chains. This the main reason is why carbon the basis for life on Earth.
The reason carbon can make long chains is because it has four open valence electron spots, so it can bind with four other elements all at the same time. Carbon combines with oxygen to form carbon monoxide or carbon dioxide, for example. Carbon dioxide is what humans breathe out during respiration. If the amount of oxygen in the room is limited or used up, it turns to carbon monoxide, which is toxic.
Carbon creates more compounds than every other element combined. There are several million different carbon compounds currently known to science. The element is required in organic chemistry, but its ability to form long bonds also shows up in other places such as diamonds, which are long crystal formations of carbon molecules.
New forms of carbon chains, such as Buckyballs, are very strong and hard to break. There are few other elements that have this property, with one of them being silicon. That is why some people believe silicon could be what aliens are made of if they are not made of carbon like all life on Earth.