Today the Accordion family includes several different instruments, all Free Reed Aerophones by definition.  Here are the main members:

Bandoneon

The Bandoneon was invented by the German Henrich Band in the 1840's and intended as a religious instrument, as opposed to the German Concertina which was considered a folk instrument.  Today, the Bandoneon is most popular in Argentina, where it was introduced by German immigrants in the early 20th century.  Bandoneons feature button keys on both sides, and most Bandoneons are diatonic, meaning each button produces a different tone on the out-bellows and in-bellows.

Button Accordion

Button Accordions were the original type of Accordions, developed from the Chinese Sheng in Europe in the 1820's, through Buschmann's Handaeoline, possibly the first instrument we could clearly recognise as the forebearer of the modern accordion.  Today, many different right hand keyboard systems exist for Button Accordions, however the most popular is a Chromatic system.

Concertina

It is considered that the Concertina was invented in England and in Germany, probably independently.  The Wheatstone Concertina was invented in England in 1829, though Wheatstone did not file for a patent until 1944.  Carl Friedrich Uhlig announced his Concertina invention in Germany in 1934.  A Concertina is different from Accordions in that the buttons are located on the ends of the bellows, and when pressed, travel in the same direction as the push/pull of the bellows. (On an Accordion, the buttons or keys are pressed perpendiular to the travel of the bellows)  Many different keyboard systems exist with Concertinas, and can be both monotonic (typically English) or diatonic (typically German).  English Concertinas are typically hexagonal in shape and have a chromatic right hand system.

Diatonic Accordion

Diatonic or bisonoric means two-toned, so a Diatonic Accordion will produce a different tone/note for each button, depending on whether the button is played on the in-bellows or the out-bellows.  Sometimes these instruments are also called Melodeon, and are limited to 1, 2 or 3 rows of buttons in the right hand.  With so few buttons, the right hand layout is usually basd around a key or two, the most common being D and G.

Piano Accordion

In 1859, French inventor Busson first fitted a three-octave piano keyboard to the Wheatstone Concertina, naming it the Organ Accordion.  However the Piano Accordion did not become popular in England until the 19th century.  Today, it is the most poular type of Accordion in New Zealand, mainly due to its easily recognisable and simple chromatic piano keyboard system.