The complicated molecules that make up living things usually contain
carbon. why is carbon so important in these molecules?

The carbon atom is capable of forming four covalent bonds in combination with other elements as hydrogen, oxygen, nitrogen, sulfur, phosphorous and halogens. The compounds formed are called organic molecules which because of the nature of the chemical bond are stable compounds.

Adding an amino group -NH₂ and a carboxylic group -COOH to a carbon chain containing carbon and hydrogen we form amino acids. Condensation of amino acids will form peptides and further proteins, which are chemical molecules from which living things are made.

Carbon is the backbone of the molecules from which life is made.