Silica is the most abundant metal oxide and the main component of the Earth's crust. Its behavior in contact with water plays a critical role in a variety of geochemical and environmental processes. Despite its key role, the details of the aqueous silica interface at the microscopic molecular level are still elusive. Here we provide such a detailed understanding of the molecular behavior of the silica-water interface, using density functional theory based molecular dynamics (DFTMD) simulations, where a consistent treatment of the electronic structure of solvent and surface is provided. We have calculated the acidity of the silanol groups at the interface directly from the DFTMD simulations, without any fitting of parameters to the experimental data. We find two types of silanol groups at the surface of quartz: out-of-plane silanols with a strong acidic character (pKa = 5.6), which consequently results in the formation of strong and short hydrogen bonds with water molecules at the interface, and in-plane silanols with a pKa of 8.5, forming weak hydrogen bonds with the interfacial water molecules. Our estimate of the quartz point of zero charge (1.0) is found in good agreement with the experimental value of 1.9. We have also shown how the silanols orientation and their hydrogen bond properties are responsible for an amphoteric behavior of the surface. A detailed analysis has identified two species of adsorbed water molecules at the solid-liquid interface, which using the language of vibrational spectroscopy can be identified as "liquid-like" and "ice-like" water or, in other words, water molecules forming respectively weak and strong H-bonds with the oxide surface. These two populations of water are in turn responsible for two distinct peaks in the infrared spectrum of interfacial water and thus provide a molecular explanation of the experimental sum frequency generation spectrum recorded in the literature. In the specific case of quartz, we show that the liquid-/ice-like behavior is the result of the silanol groups ability to donate or accept hydrogen bonds with different strengths, which consequently modulates the vibrational properties of the adsorbed water layer.