What are the different types of spiral galaxies besides barred?

The spiral galaxies are split into two classes, quite simply barred spirals and just spirals (or non barred). In effect a barred spiral galaxy is one which has well defined spiral arms but a large "bar" through the centre. A "normal" spiral galaxy does not have this bar feature.

Above is a diagram of the Hubble tuning fork which is a classification scheme for galaxies. The image gives us a good idea of the shape of the galaxy and its classification. As can be seen the spirals are broadly split between barred and non-barred.

In general, galaxies are typically divided into the barred spirals, spirals and elliptical. The "S0s" are called lenticular galaxies which are ones that sit between elliptical and spiral.

Another class which does not fit the tuning fork are the irregular galaxies, these are usually satellites galaxies (small galaxies which orbit larger ones) and don't usually hold an elliptical or spiral shape.

Two examples of non barred spirals are:
The whirlpool galaxy, and:

Messier 81 (Bode's Galaxy)

An of example of a barred spiral galaxy is:

NGC1300

Quite clearly, NGC1300 has a large bar across its center which is absent in the other two examples.

Barred spirals typically form when stars deviate from their typical circular orbit and instead end up in an elliptical orbit around the galactic center. More and more stars will gather around this orbit (locked into it most likely due to the gravitational attraction of the stars already there) thereby forming a bar. Whatever causes the initial deviation is still up for dispute.