The commonly-used galaxy classification indicates a galaxy's
morphology (galaxy morphology), i.e., shape and overall
structure, based upon its visual appearance.
Edwin Hubble developed the basic
classification still used, the Hubble types, known collectively
as the Hubble sequence or Hubble classification (though current
usage includes refinements subsequent to Hubble's work, e.g.,
by Allan Sandage and Gérard de Vaucouleurs):
elliptical galaxy: E0 through E7, the smaller the number, the closer to spherical.
spiral galaxy: S or SB (for barred) or SA (to specifically indicate unbarred) or SAB (for something between barred and unbarred, an intermediate spiral galaxy) followed by a, b, c, or d, from the tightest winding of arms to the loosest. Pairs ab, bc, and cd indicate between a and b, etc.
lenticular galaxy: SO, SAO or SBO: bright central bulge in disk but no visible spiral, the A and B for unbarred and barred.
irregular galaxy (or subsequently described as amorphous): Irr, Im, or sometimes just I.
(The term disk galaxy for a disk-like shape includes the above
spiral and lenticular galaxies.) Hubble laid out the types in
a Y-like diagram in the shape of a horizontal tuning fork and you
do see references to the classification-structure as a tuning fork.
Among the improvements over time have been suffixes and some prefixes,
to specify more detail, often describing galaxies formerly classified
as "irregular":
m suffix to SA or SB - spiral but irregular, such as Large Magellanic Cloud (LMC), which is classified as "SBm", "m" for "Magellanic type spiral".
Parenthesis around suffix letters appear to be a means of distinguishing
them, e.g., the parenthesis in "(s)m" makes clear that there isn't
some specific meaning to the pair, "sm".
Dwarf galaxies have their own terms: dE,
dS, and dIrr for dwarf elliptical, spiral, and irregular,
and dSph for dwarf spheroidal galaxy.
Some example galaxies/types:
Another classification system is the Morgan classification,
and one of its classifications, cD, is still commonly seen
for central elliptical galaxies. There also exists some classification
specific to radio galaxies.