Cinder cone volcanoes are what most of us think about when we hear the word "volcano". It has a normal vent, and then has a bowl shaped depression at the top that holds lava that flows up from the vent.
Composite volcanoes are also sometimes called stratovolcanoes. They are shaped like a normal mountain, and sometimes have multiple vents leading underground. When they have multiple vents, it means that they have different openings to the lava chamber, and when they erupt, they sometimes cause multiple reactions firing off other eruptions.
Sheild volcanoes are the largest volcanoes on Earth that actually look like volcanoes. They are made up of 90% fluid lava, this means that there is not a whole lot of pyroclastic material in the eruption. They are abundant across Hawaii, and can be about three to four miles wide at the base.
Lava domes are barely volcanoes at all, all they are is slow moving lava, not pressurized enough to do any harm, or escape the vent for that matter. Large masses of lava come out and dry at the top of the vent, and just sit there. These volcanoes are reletively small in size. The only time they actually do damage is when they expand to far and the lava gives and it blows out the top, and sometimes the sides.
You have probably never heard of this type of volcano. That is because the last time one erupted was 83 AD. These are the most dangerous volcano imaginable. When they erupt, it is almost like an explosion, they have multiple vents and unlike composites with multiple vents, when they go off, it is only a matter of time until the next vent opens, there is no "maybe". These are also dangerous, because they store up and pressurize magma, and when it erupts, it could keep spewing lava for days, even weeks. An example of this type of volcano is Yellowstone.
This is basically a field of mini volcanoes, none of them large enough to create big eruptions. They are made up of a bunch (usually 100-1000) of vents that are connected to the magma below.