I am doing a report and I need to know and FAST!
Many ways but they can be grouped into igneous methods and hydrothermal methods. Igneous -- that is rocks formed from a liquid. A liquid rock (typically at a temperature of 600-1000°C) crystallizes various minerals. In order to form quartz (which is SiO2), it must have a high silica content. Basalts (Iceland or Hawaii, for example) do not crystallize quartz as they have about 50% SiO2. But granitic liquids (say 70-75% SiO2) can crystallize quartz. So granites are quartz + feldspar + other minerals. Hydrothermal -- silica is not very soluble in water but if the pressure is high and the temperature is high, SiO2 dissolves. In fact, at 10,000 bars and 1000°C, quartz and water have complete mutual solubility. If a hot fluid rises and cools, the solubility of silica forms and quartz crystallizes. In this case, you get veins of quartz -- sometimes pure quartz which you can never get by igneous methods. Silica cements are similarly crystallized in sediments but they do not really become quartz until they are cooked. Beach sands are dominantly quartz but that is quartz that is derived from the sources described above but it is very hard and so hard to break down. Quartz-bearing sedimentary rocks are buried and can be heated and melted -- and make granites so round and round we go.
