Agaricus bisporus is a white mushroom now widely cultivated. It is the default ‘mushroom’ common in shops. In France the cultivated version is known as 'champignon de Paris'. They come in many forms. In the UK the word 'mushroom' alone most frequently refers to these.
Button mushrooms are a ball-shaped, immature form of cultivated mushroom, harvested before the gills are exposed and which mature to an open cap style and then to flat mushrooms.
There are brown and white forms, the brown being called brown mushrooms or chestnut mushrooms and, more recently, crimini mushrooms. A clever marketing strategy has been to give them Italian sounding names, including portobellini for small brown button mushrooms through to portobello or portabella for the larger brown mushrooms. Nonetheless, despite the numerous names, they are all forms of the basic cultivated mushroom.
All these mushrooms, whilst lacking some of the punchy flavour of various wild mushrooms, are good, adaptable additions to stews and sauces, fried or grilled, and, with the addition of a handful of reconstituted dried porcini, can make a great soup.
(If gathering mushrooms you must be absolutely certain what you have before you eat them as many are very poisonous.)