What is the difference between an accordion,a piano accordion and a concertina ?  

A piano accordion has piano style keys on the right hand side (and bass and chord buttons on the left). It's a chromatic instrument which can play in any key. It is often just referred to as an 'accordion'.

A melodeon (which is sometimes called an 'accordeon' with an 'E', especially in Ireland) is an instrument, shaped more or less like a piano accordion, with one, two or three rows of buttons on the right hand side. These buttons play different notes on the 'push' and 'pull' strokes, rather like a harmonica/mouth organ. Each of the right hand rows is diatonic (can play only in one key) but the three rows may be arranged so that all of the chromatic scale is available.

A concertina is an instrument which is usually either hexagonal or octagonal in section. It's a smaller instrument than the accordion or melodeon and there is no keyboard sticking out (as in the piano accordion). The buttons are arranged on the ends of the instrument and one usually needs a strap across the back of each hand (or a thumb strap on some models) so that you can push and pull. There are two basic kinds: the English concertina (invented by Wheatstone who also invented the Wheatstone telegraph), where each button plays the same note in and out, and the Anglo-German concertina which is rather similar to the melodeon in the way it is played - it is diatonic and has a different note on the push and pull strokes.

There are other similar instruments which are not so often seen. The Russian accordion is like (and sounds very like) a piano accordion but has about five rows of buttons on the melody (right hand side) and a very complicated fingering system. A similar instrument was used in French cafe music. People who play them say that they give a quite different expression to the music than the piano accordion.

The other instrument which is popular in tango music is the bandoneon. This is an oblong box but with buttons on the ends, rather than a keyboard. It is a very flexible instrument and quite soft toned. A great exponent (and writer) for the bandoneon was Astor Piazzola. His music uses the tango style but raised it from a dance style to the realms of classical music. There are videos of him and other performers on YouTube which are truly breathtaking.