Most galaxies show one of a set of predictable forms, typically falling into the categories of elliptical, spiral or barred spiral. Some 20-25% of galaxies, however, fail to show such a structure, appearing instead in a wide variety of random or chaotic forms. These are irregular galaxies, placed in their own 'Irr' category on the Hubble Classification.
In most cases, these irregular shapes are due to historical interactions between galaxies, whereby the gravitational forces of another galaxy have stretched and deformed the original shape, breaking its structured pattern and leaving a relatively formless mass of material. Interactions like this will sometimes leave hints of the original galactic form intact, and can also at times leave 'tidal tails' - telltale streams of galactic matter torn out by another galaxy's gravity and left extending out into intergalactic space.
The reorganisation of matter within an irregular galaxy means that at times greater concentrations of gas can collect together than in a typical galactic form, gathering the raw material for star formation into far greater densities than is typical. In cases like these, the result is the emergence of brilliant eruptions of new stars, giving rise to a phenomenon known as a starburst galaxy.
The closest irregular galaxies to the Milky Way are the two Magellanic Clouds, both of which show signs of originally having a barred spiral shape before the Milky Way's gravity broke apart their structured shape (indeed, the Large Magellanic Cloud is officially classified as a barred spiral galaxy, though that form is difficult to distinguish). Other prominent examples of irregular galaxies are M82 in Ursa Major (an irregular of the starburst type) and IC 10 in Cassiopeia.
Some smaller irregular galaxies have a different origin. These are the Dwarf Irregular Galaxies (abbreviated to dIs or DIGs): relatively young, primitive galaxies with low metallicity (showing that their stars have yet to manufacture more complex elements in significant quantities). Examples of these dwarf irregulars include NGC 1156 in Aries, or the 'Peg DIG' (Pegasus Dwarf Irregular Galaxy) within the Local Group.