Humanity is a measure of how closely a vampire clings to the morality and values of her life as a mortal human being, and consequently how well they are able to resist the urges of the Beast.
Since every Kindred was a human before their Embrace, their most natural response in resisting the Beast's feral, predatory nature is to cling to their humanity. This is generally represented by the strength of their conscience and self-control, virtues common to Kindred and mortals alike.
A Kindred's humanity has a direct effect on the strength of the vampiric curse; those who lose ground to the Beast and lose their humanity find it more difficult to interact with mortals, to be active during daylight hours, and to awaken from long periods of torpor.
Humanity is universally adopted by the vampires of the Camarilla, but even among the Sabbat it is the "natural" and most common way of dealing with the Beast. Some vampires however reject the mores of mortal existence and adopt an alien Path of Enlightenment, in which they learn or construct a wholly different standard of morality. These often incorporate different virtues - conviction instead of conscience, and instinct instead of self-control - and establish a very different relationship with the Beast. Replacing the morality of one's human life is a daunting task, not to be taken lightly, and generally impossible without the tutition of one who already follows the alternate path.
The practice of assuming a Path of Enlightenment was much more common in the medieval period, when every vampire had to choose a "Road", a quasi-religious path to harmony with or control over the Beast. Humanity was one of these Roads, known then as the Road of Humanity (Via Humanitas). Though less common than today, the Road of Humanity was still the most natural Road to follow.