What is the name of the bond that holds complementary bases together?

The bond that holds complementary DNA bases together is a hydrogen bond. These non-covalent attractions connect paired bases across the two strands of the double helix.

In standard Watson–Crick pairing, adenine bonds with thymine through two hydrogen bonds, while guanine bonds with cytosine through three. The matching shapes and chemical groups of the bases make these pairings much more stable and specific than most alternative combinations.

Hydrogen bonds allow the strands to separate when DNA must be copied or transcribed, because they are individually weaker than the covalent bonds forming each sugar-phosphate backbone. This gives DNA both stability and accessibility. The backbone itself is held together by strong phosphodiester covalent bonds, so it is a different part of the molecule with a different kind of linkage.

A subtle scientific detail is that base stacking interactions also contribute substantially to the overall stability of the DNA helix. Nevertheless, for the direct bond between complementary bases, “hydrogen bond” is the standard and correct answer. Peptide bonds join amino acids, not DNA bases.