Gabriel: A key here is that hydrogen bonds are relatively weak interactions BETWEEN molecules. So a COVALENT bond between an oxygen and hydrogen in one water molecule, for example, is NOT a hydrogen bond.
The origin of hydrogen bonding is the unequal 'sharing' of electrons between a hydrogen (with a weak 'pull' on electrons, more formally a relatively low electronegativity) and generally an oxygen or nitrogen that the H is covalently bonded TO. Another way of looking at it is:
H--->O<---H, where the arrowheads indicate the Oxygen is 'winning' the tug-of-electrons.
This leaves the HYDROGEN somewhat positive overall (because it's 'lost' its fair share of electrons) and the oxygen correspondingly negative.
A hydrogen bond is the interaction that results between the somewhat-postive H from ONE molecule and the somewhat-negative O or N on ANOTHER.
Note that carbon also has a relatively weak electron pull, so oxygens & nitrogens that are covalently bonded to carbons are also somewhat negative in charge and can participate in H-bonds on other molecules.
If you look at HOW two bases in DNA or RNA are interacting, you should be able to see the H-bonds! (H-bonds are often shown as dashed lines to indicate they are different & weaker than covalent bonds)