A mountain is basically any part of the crust that juts out to high elevation, with steep slopes. As opposed to a plateau, which is a location of generally high elevation.
A volcano is a place on the Earth's surface where magma erupts i.e. where volcanism occurs. Because the magma generally comes to rest (in whatever form) somewhere in the vicinity of the eruption site, it tends to form a pile that rises over the ambient landscape and we call this pile of volcanic rocks a volcano. If it gets to be high enough, we might also call it a mountain. What 'high enough' means is quite arbitrary though.
Not all volcanoes rise higher than the surrounding landscape. Particularly violent eruptions can remove so much material from a magma chamber that its top collapses and leaves a hole in the landscape. Lake Taupo in New Zealand is an example.
