This hydrogen bonding is way different from other uses from the word "bond", since it is a force of attraction between a hydrogen atom in one molecule and a small atom of high electronegativity in another molecule. This means that it is an intermolecular force, not an intramolecular force as in the common use of the word "bond.
When hydrogen atoms are joined in a polar covalent bond with a small atom of high electronegativity like O, F, or N, the positive charge on the hydrogen is highly concentrated because of it's small size, this means the more the less concentration. If the hydrogen is close to another oxygen, fluorine, of nitrogen in another molecule, then there is a force of attraction called a dipole-dipole interaction. This attraction or now called "hydrogen bond" can have about 5 percent to 10 percent of the strenght of a covalent bond.
Hydrogen bonding has a very important effect on the properties of water and ice.
Hydrogen bonding is also very important in proteins, nucleic acids, and also in life processes. The opening of DNA is a breaking of hydrogen bonds which help hold the two strands of the double helix together. This is how hydrogen bonding is very important to us. For attraction to create a hydrogen bond the hydrogen must become extremely positive and the other aom must become negative to combine a bond, "hydrogen bond".