Barred galaxies per se do not exist. They are just a sub category of spiral galaxies of which about 2/3rds are barred spiral galaxies.
It's not known to any degree of accuracy, but about 66% of all spiral galaxies are barred and about 60% of galaxies in the local Universe are spiral galaxies.
the two types of galaxies that are barred and normal would be irregular as my best guess
They are called Spiral Galaxies or Barred Galaxies.
Spiral galaxies, barred spiral galaxies, elliptical galaxies, and irregular galaxies
Barred galaxies per se do not exist. They are just a sub category of spiral galaxies of which about 2/3rds are barred spiral galaxies. A barred spiral galaxy has a central bar shaped structure composed of stars whereas a non barred spiral galaxy does not. See related link for a pictorial of the difference.
the types of galaxies are barred, irreagular, spiral, and a shape like an hourglass.
spiral, eliptical, cartwheel, barred spiral, and lenticular galaxies
The type of galaxy that is Barred and Normal is the Spiral Galaxy.
The two Magellanic Clouds are irregular dwarf galaxies.
The bar.
We believe that the Milky Way is a barred spiral galaxy. Follow the link to see Hubble photos of other barred spiral galaxies.