Dictionary Definition: 

“A weak bond between two molecules resulting from an electrostatic attraction between a proton in one molecule and an electronegative atom in the other.”

Selected from OxfordDictionaries.com

Functional Definition:

Hydrogen bonding is one of the first types of nonconvalent  bonding we come across in introductory chemistry. In hydrogen bonds there is a weak intermolecular attraction between an area of partial negativity and an area of partial positivity. The partially negative region of the attracting molecule usually has high electronegativity such as that of oxygen. The partially positive region is usually a proton (-H).

The most common example given for hydrogen bonding is water, where the oxygen of one molecule slightly attracts the hydrogen of another. This web of weak attraction is what characterizes water; high surface tension and high freezing and boiling points. In plants a model known as the Cohesion-Tension Theory suggests that the weak hydrogen bonds of water in the xylem compound on one another creating an overall pull from root to the stomata of water as it transpires from the leaves. This pull causes the against-gravity flow of water in plants as small as grasses and as large as the great Redwoods. Hydrogen bonding is also what makes DNA so strong and yet allows it to be unwound and copied with relative ease. The a-chitin of arthropods and fungi is also stabilized by hydrogen bonding.