In Watson's like-with-like model of DNA, the base pairs are held together by hydrogen bonds.
Hydrogen forms covalent bonds with other molecules to produce molecules which are more stable than the free atoms. The electron from the hydrogen is effectively 'shared' between the hydrogen and the atom to which it has bonded. Another electron, from the bonded atom, is also shared between the two.
However, if the hydrogen is bonded to an oxygen or nitrogen atom the sharing of the electrons is not quite equal. The bond becomes slightly polarised. This means that the hydrogen atom has a slight positive charge while the oxygen or nitrogen atoms are slightly negative.
If two molecules with this polarisation are close to one another they can 'hydrogen bond'. The positive hydrogen atom of the first molecule is attracted to the negative oxygen or nitrogen atom of the second molecule. This bond is weaker than the covalent bond, but is the attraction which gives DNA its shape.