Any rock that solidifies from lava is a volcanic rock. All volcanic rocks are igneous rocks, and are perhaps more correctly known as extrusive igneous rocks; this is in contrast to magma that solidifies beneath the ground, which is known as plutonic rock or intrusive igneous rock. Volcanic rocks, particularly basalt, makes up much of the Earth's crust, and nearly all of the ocean floor.
There are a number of ways that you might classify volcanic rocks; the most important classifications are based on the minerals contained in the rock. The type of rock formed from lava depends on the chemical composition of the magma from which is comes. Magma is graded along a continuum of mafic to felsic. Mafic rocks have comparatively low silica content, and therefor solidify more slowly, and are less viscous lavas. Felsic rocks have lots of silica, and therefor harden quickly and flow more slowly.
||Calcic plagioclase feldspar, pyroxene, and olivine.
||Basalt flows easily, and can cover a lot of ground. Because of this it is the most common extrusive igneous rock. It is particularly common at divergent plate boundaries, and thus the sea floor is made primarily of basalt. It is very dark in color.
|| Plagioclase, pyroxene, and hornblende.
||Common in subduction zones, and therefor often found in mountain ranges. Andesite contains lots of phenocrysts, and thus classification of andesite often depends on the most abundant phenocryst.
|| Plagioclase feldspar, biotite, hornblende, and pyroxene.
||Formed when young oceanic crust is subducted. Dacite often forms as an intrusive formation in sills and dikes, and often erupts explosively in its volcanic form.
|| Alkali feldspar.
|| Typically porphyritic. It crystallizes quickly, staying close to the volcanic vent. Trachyte often forms from fractional crystallization of basaltic magma at high pressure, as the felsic minerals crystallize out of the magma and form a comparatively mafic magma.
||Quartz, alkali feldspar, biotite, and hornblende.
||Very viscous, solidifying quickly. It is comparatively rare, but may be present in large masses where it does appear. Often pinkish-gray in color.
||Usually the same as rhyolite.
||Not technically a mineral, and therefor not technically a rock. Obsidian forms when felsic lava cools so quickly that that it does not have time to form crystals. Obsidian is technically a glass, not a rock, but geologists, being a liberal group, have consented to have it grouped with the igneous rocks.
Once the lava hardens, it can create a large range of formations. As noted above, the silica content (along with temperature) determines the rate of flow of the lava, but it also helps determine the shape of the resulting volcanic cone and the physical structure of the resulting rocks.
(low silica content)
(high silica content)
Fluid, fast-moving basaltic lavas forming Pahoehoe and a'a type lava flows. The resulting cones are the low-profile shield volcanoes. Felsic lava forms into basalt, which is the most common type of volcanic rock worldwide. Pahoehoe may make interesting formations, such as ropey folds and lava tubes. Pillow lava and hyaloclastite are also generally mafic.
Andesite domes and block lavas formed from basaltic andesites. It is intermediate in all ways; between the mafic and felsic lavas in flow rate, temperature, and explosiveness.
Forms lava spines and lava domes, formed by lava that solidifies shortly after leaving the vent. Even though these lavas solidify comparatively quickly, silica-rich rhyolitic lava flows can cover tens of kilometers. Felsic lava is the most explosive, as rapidly forming silica bonds can trap gases which then escape violently, resulting in rocks like pumice and reticulite
Lava with particularly high silica content may form glassy Obsidian, including interesting formations like Limu o Pele and Pele's hair.
Volcanic rocks aren't all formed from lava flows; some are formed from explosive eruptions, resulting in dust, pebbles, and boulders ejected from the vent. These are called tephra, or in their consolidated forms, pyroclastic rocks. Pyroclastic flows, nuee ardentes, and ash clouds are often the most spectacular and damaging result of a volcanic eruption, and looking at the rocks blown free from an explosion can tell us a lot about the force of the volcano.
||2 mm or less.
||Exploding gases, thermal shock, and entrainment of highly fractured surface rock result in a tremendous amount of dust and ash, which can form clouds that last for months, and can travel for hundreds of miles.
||Tuff. This is sometimes lumped in with tufa.
||2 to 64 mm
||Most volcanic cinders fall into this category. Highly fractured country rock may also form lapilli.
||Lapilli tuff or lapillistone.
||64 mm or greater.
||Bombs specifically refer to large clumps of lava that have become airborn. These may literally explode, either when they hit the ground (due to impact) or latter (due to expanding gases).
||64 mm or greater.
||These are masses of country rock that have been blown into the air by the force of the eruption.
There are also carbonatite volcanoes, in which carbonate rocks are melted and erupt. All the known examples are inactive but for one, Oldoinyo Lengai. This single volcano produces a thin, comparatively cool (500 c) lava composed of sodium carbonate. At least some of these volcanoes are water-soluble, so we can't even be sure how common they may have been in the past, or what formations they may have created. These carbonatite volcanoes are still an enigma, but extremely cool.
If I've forgotten anything important or interesting (or messed up the formatting of the tables!), /msg me and I'll do my best to add it in. Don't forget to follow the links, a lot of this stuff is noded in greater detail elsewhere.
Thanks to Auduster for his help.