The amount of UV which gets in will vary depending on the type and treatments of the glass and the thickness of the glass. Almost all glass (certainly all commercial glasses) absorbs some UV light.
The reason for this absorption has to due with the bonding of each glass type. Most commercial glasses are sodium-silicates. These glasses consist of a connected silica network, which is broken up by an alkali (sodium). The addition of sodium breaks the network creating non-bridging oxygens, which lowers the melting temperature as well as reducing the energy of electromagnetic radiation needed for absorption. In combination with small amounts of iron impuriity, which leads to intense UV absorption bands, these non-bridging oxygens prevent the transmission of UV light.
It should be noted that different glasses have different UV edges (the frequency of UV light that begins being absorbed). Borosilicate glass with small amounts of alkali will actually transmit further into the UV spectrum as a result of the boron anomoly, with fused silica (pure SiO2) glass transmitting the furthest. Even fused silica though will only transmit into the very low energy UV light.
In essence, the more loosely the electrons are bound, usually because they are not involved in covalent bonding in the case of non-bridging oxygens, the less UV light is transmitted.
The process of photochromitism utilizes this effect where charge transfer bands from added transition metals cause glass to change color when exposed to UV light. This is the result of electron transitions within the glass to more excited states. When the glass is no longer exposed to UV light the electrons return to their equilibrium position and the glass clears.
Most tempered glass, i.e. door glasses, back glasses, quarter glass...is UV glass.
Quartz will pass most UV.