An eruption is a sudden outpouring of ash, gas and or lava from a volcano.
Different types of volcano erupt in different ways. If the volcano comes out of a large gash in the Earth's crust, as it often does in Iceland, for example, then the eruption will be mostly runny lava. The volcanoes on Pacific islands also mostly erupt lava. The ones on Hawaii are shown in the video below. The lava cools to a black rock called basalt.
In other parts of the world, volcanoes occur because two parts of the earth's crust are pushing together, causing the crust to break up and allow liquid rock (magma) to esape to the surface. In these cases the lava is much more sticky and usually does not flow out of the volcano, but gets blown out by the pressure of gases inside it. As the lava is blown apart it cools very quickly to give ash. Eruptions of this kind do not happen very often, and they are unpredictable and so dangerous. Volcanoes all around the outside of the Pacific Ocean (the Pacific Rim of Fire) are like this.