Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
Because of this fact, invertible sheaves are closely related to line bundles, to the point where the two are sometimes conflated.
Let X be an affine scheme Spec R. Then an invertible sheaf on X is the sheaf associated to a rank one projective module over R. For example, this includes fractional ideals of algebraic number fields, since these are rank one projective modules over the rings of integers of the number field.
Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product.
The direct construction of invertible sheaves by means of data on X leads to the concept of Cartier divisor.