What is the geometric name for plane shapes which are like polygons but made of concave circular arcs? If a number of circles form a ring around a centre with neighbouring circles touching, these are the shapes of the areas between them.

I believe the term you are looking for is Reuleaux polygon. According to Wikipedia, "In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constant radius."

https://en.wikipedia.org/wiki/Reuleaux_polygon

I don't think that is the shape, as a Reuleaux polygon has convex sides formed when circles overlap. The shape formed by circles that are mutual tangents to each other (as described in the question) has concave sides, since the circles that form it have their centres outside the shape. There is a picture here that has the relevant shape in its centre, although it is obscured because of the subsequent addition of more circles of different sizes.

https://en.wikipedia.org/wiki/Apollonian_gasket

Never mind, it looks like I found it.
The four cusped curve, for example, those shapes on the Pittsburgh Steelers logo are called astroids (like asteroids but without an E).
The general case with 4 or other numbers of cusps are called hypocycloids. I can't say I'm 100% sure that the curves in a hypocycloid are actually circular arcs, but this is near as dammit to what I was looking for.
https://en.wikipedia.org/wiki/Hypocycloid