The heyday of the concertina has to be the late 19th and early 20th centuries. They are conceived and designed to be portable and easy to play. Like the harmonica which was also invented in the first half of the 19th century concertinas were designed for times when people had to entertain themselves.
Being essentially a folk instrument, the concertina has a diatonic scale, in more common parlance that is a range of notes based around the notes of a particular major scale rather than having the full chromatic range needed to play most kinds of western music in all keys.
The intention is to be able to knock out popular tunes and the simple 20 button Anglo model has essentially one set of 10 buttons in the key of C split evenly between the left and right hands and another set of 10 buttons in the key of G also split evenly between the left and right hands.
The concertina falls into a musical instrument family known as open reed instruments. Others in this broad category would be the accordion, the harmonium and bagpipes. I know statement of the obvious, but whereas the bagpipes have a bag that the player fills through a blow pipe with a one-way valve and by squeezing the bag maintains constant pressure and a constant flow of air over the pipes with the reeds, all the other instruments mentioned here including the concertina have hand operated bellows.
Concertinas and most button accordions are distinguished by playing different notes on the bellows push vs the bellows pull for any button or buttons depressed while the piano accordion and harmonium play the same notes when the same keys are depressed on both the bellow push and the bellow pull. There are proper fancy expressions for that, unisonoric for making the same note and bisonoric for making different notes but I prefer my convoluted confusing explanation any day.
The first decades of the 19th century were a period of experimentation and a number of craftsmen across Europe were turning out a variety of accordions and concertinas with different combinations and numbers of buttons designed to allow a solo player to produce a melody line and accompaniment at the same time.
After a few decades the concertina market settled into two broad families – the Anglo 20 and 30 button versions patented and produced by Sir Charles Wheatstone in England and a German concertina developed by Carl Friedrich Uhlig also with different numbers of buttons. These are the two families of concertina that survived to the present day.
The button system on the Anglo and the German versions are so different that a player used to one version would have a hard time switching to the other version. I won’t try and be any more exhaustive than that. Much more on the history and development can be found on Wikipedia.