Phenol is able to lose a hydrogen ion because the resulting negative charge formed at the oxygen atom is stabilized by delocalizing around the benzene ring. As a result, phenol is weakly acidic, whereas most other alcohols do not have this property.
Phenol is a benzene ring with one hydrogen atom replaced with a hydroxyl (OH-) group. Benzene rings are often depicted as having alternate double and single bonds, but in actuality, all the bonds have the same length, which is shorter than a typical single bond, but longer than a typical double bond. This is because the pi electrons are actually delocalized across the entire ring. In other words, there are no localized single or double bonds between atoms. Instead, the valence electrons are able to travel around the whole ring.
This property of benzene is what gives phenol the properties of a weak acid. The hydroxyl group is able to lose the proton (H+) because the resulting negative charge that would normally reside over the oxygen atom is able to spread across the entire ring, thus stabilizing the resulting ion.
Conversely, most alcohols do not have this ability. If the hydrogen were to leave a functional alcohol group on other compounds, the resulting negative charge would lie solely over the oxygen atom. This is not energetically stable, and therefore, the hydrogen is unlikely to separate from its original compound. Thus their acidic properties are so weak, they are virtually ignored.