We investigate a class of rapidly growing emission line galaxies, known as 'Green Peas', first noted by volunteers in the Galaxy Zoo project because of their peculiar bright green colour and small size, unresolved in Sloan Digital Sky Survey imaging. Their appearance is due to very strong optical emission lines, namely [O iii] lambda 5007 A, with an unusually large equivalent width of up to similar to 1000 A. We discuss a well-defined sample of 251 colour-selected objects, most of which are strongly star forming, although there are some active galactic nuclei interlopers including eight newly discovered narrow-line Seyfert 1 galaxies. The star-forming Peas are low-mass galaxies (M similar to 108.5-1010 M(circle dot)) with high star formation rates (similar to 10 M(circle dot) yr-1), low metallicities (log[O/H] + 12 similar to 8.7) and low reddening [E(B - V) < 0.25] and they reside in low-density environments. They have some of the highest specific star formation rates (up to similar to 10-8 yr-1) seen in the local Universe, yielding doubling times for their stellar mass of hundreds of Myr. The few star-forming Peas with Hubble Space Telescope imaging appear to have several clumps of bright star-forming regions and low surface density features that may indicate recent or ongoing mergers. The Peas are similar in size, mass, luminosity and metallicity to luminous blue compact galaxies. They are also similar to high-redshift ultraviolet-luminous galaxies, e.g. Lyman-break galaxies and Ly alpha emitters, and therefore provide a local laboratory with which to study the extreme star formation processes that occur in high-redshift galaxies. Studying starbursting galaxies as a function of redshift is essential to understanding the build up of stellar mass in the Universe.