It depends on which type of volcano that you are asking about. Volcanoes come in three types: non-explosive, explosive, and composite (strato) volcanoes. I'll go into a bit of detail in each. They all start erupting in basically the same way, with the refilling of an empty magma chamber.
Beneath each volcano is a magma chamber, which is left empty after an eruption. Over the course of time, the magma chamber refills with magma. Once it is full, additional magma still tries to squeeze in there. It can't though since the chamber is at maximum capacity, so pressure begins to build up. The magma is squeezed and squeezed, then starts making its way up through the Earth's crust through various vents leading out of the chamber and through the crust. Eventually, the magma finds a vent that leads to the surface or the magma simply blasts through the surface. In either case, a volcanic eruption occurs.
A non-explosive eruption is characterized by large lava flows and very little to no pyroclastic material. The magma that correlates with the eruption is usually low in silica, which makes the lava more "runny." Kilauea volcano in Hawaii is this type of volcano.
The complete opposite of a non-explosive eruption is an explosive eruption. The magma propelling an explosive eruption is typically very high in silica, which makes the magma thick and goopy. It tends to quickly plug vents. The resulting pressure behind those plugged vents causes the eruption to explode and blast out large amounts of pyroclastic material. There might be some lava, but often not. Pericutin volcano in Mexico is a good example of this type of volcano.
The last volcano type is the composite volcano. It's a combination of the two previously discussed. During a composite eruption, the composite volcano usually erupts in a big explosion with lots of pyroclastic material. That explosion clears the volcano's "throat," which allows for outpourings of lava. Mt. St. Helens is a good example of a composite volcano.