Extrusive igneous rocks are fine grained due to fast cooling. They cool at a faster rate than most igneous rocks.
If the rocks are igneous then in general:- Coarse = slowly. Fine = fast. However if the rocks are metamorphic or sedimentary then grain size has nothing to do with cooling and indicates other things.
Basalt
yes because of the fast cooling rate
igneous rocks
Fast cooling lava or magma creates an aphanitic or small crystal (grain) textured igneous rock.
When igneous rocks form by cooling quickly, their microstructure includes fine grains. It is the aphanitic igneous rock that includes andesite, basalt, and rhyolite. Wikipedia has more information, and a link is provided below. If you wish to wade on into the depths of rock microstructures, you can link on from there.
A coarse grained texture, referred to as a phaneritic texture, will be the resultant igneous rock texture. The slower the magma cools the more time minerals have to crystallize and thus grow bigger.
Extrusive
The difference between an Extrusive and Intrusive igneous rock is the way in which they cool. An Extrusive igneous rock cools very fast on the surface and is created by lava. Since the cooling process is very fast extrusive igneous rocks have very small crystals (fine grained). On the other hand an Intrusive igneous rock cools very slowly beneath the surface and is created by magma. Since the cooling process is very slow intrusive igneous rocks have very large crystals (coarse grained). In some instances there is also a third type of igneous rock. Technically it is an extrusive rock, but it resembles glass. This glass type of igneous rock forms when magma/lava is instantly cooled.
Igneous rock is formed from the cooling of magma or lava.
yes because of fast cooling rate
The texture is mostly dependent on the amount of time the magma or lava had to crystallize into a solid rock. Volcanic rock is formed from quickly cooling lava (also called extrusive igneous rock), and will have a fine-grained texture. Plutonic rock (also called intrusive igneous rock) is formed from slow cooling magma, and will exhibit visible crystals.