Nucleotides are held together with phosphodiester bonds - these are covalent bonds formed between the 5' phosphate group of one nucleotide with the 3' OH group of another. This way you get the phosphate-sugar-phosphate-sugar-phosphate... in the DNA backbone.
Between the two strands of the helix the pairs of nucleotides are held together with weak hydrogen bonds. There are 2 weak H bonds between A and T; and 3 weak H bonds between C and G.
