DNA is a large molecule and therefore is a suitable molecule for storing genetic information. DNA is like a code with instructions that has the ability to not only store information, but command other cells, and replicate itself. It also has the ability to build more components of cells. The DNA parts that carry this genetic information are called genes.
Base pairings are important because they control the qualitative value of the genome and phenome. What this means is that, if you pair the bases with the wrong base, a mutation can easily occur, or a disease may be formed and even passed on through genetics from generation to generation.
I will answer your two questions in reverse, because it makes more sense that way.
The base pairings are important, because each 3 base sequence gets transcribed into a 3 base Codon on an mRNA molecule. This mRNA molecule leaves the nucleus of the cell and travels to a ribosome. The ribosome reads each Codon and pairs it up with the Anticodon of a specific tRNA molecule. For eacmple:
DNA: ACG --> mRNA: UGC ---> tRNA: ACG
Each specific tRNA molecule (in this case, ACG) carries a specific amino acid (in this case, threonine). As the mRNA molecule is thus translated by the ribosome, a long chain of amino acids, arranged in a specific order, is synthesized. This chain of amino acids, called a polypeptide, will then be transported to the Golgi Body where it is processed and will become part of a protein. And proteins are used by your body to build structures (hair, muscle, hemoglobin, etc) and enzymes (which regulate all the metabolic processes).
So, the short answer: DNA stores the information used to build proteins. The specific base pairings code for specific amino acids that build these specific proteins. If you have the wrong base pairing, you will most likely get the wrong amino acid, and thus the wrong protein. This is called a... mutation!