There are three types of aster family flower heads, named according to the type(s) of florets present.

If both types of florets (ray or ligulate and disk) are present in a head, the arrangement is termed radiate.

Verbesina alternifolia
(wingstem)

If a head contains rays only, it is called ligulate.

Hieracium venosum
(rattlesnake weed)

And if only disk florets are present, a head is called discoid.

Vernonia noveboracensis
(New York ironweed)

Another characteristic of aster flower heads is the presence of phyllaries.

Symphyotrichum species, possibly S. oblongifolium 
(aster)

At the base of a flowerhead there’s a whorl of bracts (modified leaves), called an involucre. In aster family plants only, the individual leaf-like structures are called phyllaries. The number of phyllaries, how they’re arranged (how many rows), the shape, presence of hairs, and other fine details are characteristics used to distinguish certain species from others [see this recent post about tickseed].

Symphyotrichum species
(aster)

next: aster fruits/seeds