Phosphodiester bonds hold the backbone together while hydrogen bonds attribute to the double helix structure because they are the attachements between the nucleotide bases--think of a ladder, the two sides (vertical) are the backbone, which are held from top to bottom by phosphodiester bonds. Then think of the individual steps (horizontal of the ladder) imagine that they aren't connected in the center on one side there is a purine (A,G) and on the other side is a pyridine (C,T) -- you can't climb up the ladder unless the two sides are connected and in DNA's case they are connected by hydrogen bonds-- 3 H bonds between C and G and 2 between A and T. This is grossly oversimplified, but this analogy presented by my intro bio teacher has always really helped me answer these type of questions. A quick wikipedia search of phosphodiester bonds will clear this up in scientific terms.
