All organic compounds contain carbon and usually hydrogen. Oxygen, nitrogen, sulfur and phosphorous are also common in organic compounds. The reason carbon is the key atom to living things is that it's present in most of the molecules that make up lving things. Ninety percent of the nearly 10 million compounds known to exist are organic.
Carbon forms so many compounds because of its bonding ability. A single carbon atom has four valence electrons and has relatively low electronegativity so it forms four covalent bonds. It can form single, double and triple bonds, allowing the formation of many lengths of carbon chains and carbon rings that are very stable. A carbon within a ring or chain can bond to a group of atoms called a functional group to produce a particular type of compound such as an alcohol, an organic acid, or an amino acid. Fats, carbohydrates, proteins and nucleic acids are all composed primarily of carbon.
Organic compounds, by definition, contain carbon. Hence, carbon atoms are the essential component of all the organic compounds on planet Earth. The majority of organic compounds also contain hydrogen, so many texts define organic compounds as those containing carbon and hydrogen (although exceptions such as CCl4 exist).
Carbon is the key atom to all living things for several reasons. For one, we are made up of organic compounds, our bodies are organic, i.e., contains carbon. The food that we survive on is organic in nature (whether we are talking about vegetables & fruits or meat, etc.). Most of our fuels (including, coal, wood, gasoline, diesel, kerosene, etc.) are organic and contain carbon. The medicines that we consume are organic in nature. Steel, which we use in construction, automobiles, etc., also contains carbon. There are several other uses of carbon and hence it is a key atom to all living things on Earth.
Hope this helps.