A fairy ring is simply mycelium doing what mycelium does. It starts from a single point, exhausts the nutrients at the centre, and spreads outward as an expanding circle. The fruiting bodies appear at the growing edge, which is why you get a ring rather than a patch. Some rings in old grassland are hundreds of years old and tens of metres across.
The one you want
The Fairy Ring Champignon, Marasmius oreades, is the classic. Tan to buff cap with a low central bump, gills that are strikingly widely spaced and pale, and a stem that is so tough and wiry it will not snap - you can tie it in a knot. It dries on the lawn in hot weather and revives completely after rain, which is a good confirmation in itself. The caps dry beautifully and keep their flavour for years.
The one that will hurt you
Fool's Funnel, Clitocybe rivulosa, grows in exactly the same short grass and often in rings too. It is white to pale buff, the cap flattens and then funnels, and it frequently shows faint concentric watery banding. Crucially, the gills are crowded and run down the stem, and the stem snaps cleanly.
It contains muscarine. Within fifteen minutes to two hours you get profuse sweating, streaming eyes and nose, heavy salivation, blurred vision, abdominal cramps and a slowed heart rate. It is treatable in hospital with atropine, and people do recover, but it is a genuinely frightening experience and it has killed frail patients.
The rule
Crowded gills running down the stem, and a stem that snaps: leave it. Widely spaced gills and a stem that will not break: that is your champignon. And never gather from a lawn you have not walked in daylight, because the two species will grow within a metre of each other.
A third ring-former
The Trooping Funnel, Infundibulicybe geotropa, also makes large rings, usually at woodland edges rather than in mown grass. It is a big, pale, funnel-shaped mushroom with a firm stem, and it is a decent edible, but it is another species where the crowded decurrent gills mean you must be certain before it goes in the pan.