A hydrogen bonded to a very electronegative atom makes for a highly polar bond. The dipole moment favors electron density around the more electronegative atom, leaving the hydrogen with a partial positive charge. This bond is different from other bonds in the molecule because of its propensity to break into a negative ion and a positive hydrogen ion. This propensity is driven by the tendency of the more electronegative atom to take up the electrons that make up the covalent bond. In fact, one can write resonance structures for such molecule that show the bond in question already broken. The t-butyl alcohol molecule can be used again to illustrate this point.
The greatest contributor to the hybrid is obviously structure I because it is neutral. Structure II has charge separation and therefore is a minor contributor. However, the significance of structure II is that it shows the negative character of the oxygen and the positive character of the hydrogen, and therefore the polarity of this bond. A better representation of the hybrid could be structure III, which shows the oxygen with partial negative character, and the hydrogen with partial positive character.
In the Bronsted-Lowry theory of acids and bases, an acid is a hydrogen ion donor, or proton donor, and a base is a hydrogen ion acceptor, or proton acceptor. Hydrogen atoms that have a substantial degree of partial positive charge (i.e. low electron density around them) are commonly referred to as acidic protons. In the example above, the hydrogen bonded to oxygen is considered to be acidic, and the molecule as a whole is considered a Bronsted acid because it has a propensity to release a hydrogen ion, or proton.