What is the significance of adenine, thymine, guanine, and cytosine in the structure of a DNA molecule?

They make up the pairs of nitrogen bases that give DNA (deoxyribonucleic acid) its structure.

These nitrogenous bases are the digital information that structures proteins.

The Four structures give the DNA its sequence. This sequences then codes or is read by the body to form proteins. Which in turn give you your traits and features. Also the specific sequence of A(adenine) T(thymine) G(guanine) and C(cytosine) shows the genes that make up the body.

The sequence of these bases is read in groups of three at a time. Since there are four bases, and they are read in groups of three, there are 64 possible combinations in each group of three (since 4 cubed is 64).
Now, there are 20 amino acids in total, amino acids being the building blocks of protein.
So, from these 64 combinations, there are specific combinations that are used to refer to each of the 20 amino acids specificially.
(It is true that more than one base grouping may refer to any particular amino acid since there is a 64:20 ratio, but one base grouping can never refer to more than one amino acid.) Thus, each base grouping will identify a specific amino acid only. And the base groupings are read and translated into the appropriate amino acid.
Now the amino acids are strung together in the same order that the base groupings occur in in the DNA strand. This causes a specific protein to be built, and you are built out of protein.
By the way these base groupings of threes of the A, T, G, and C bases are called codons.