What is the difference between definite and indefinite integrals?

Definite integrals produces a numerical answer. It is asking the question: what is the area under the curve f(x) between x=a and x=b.

Indefinite integrals produces a function. It is asking the question: what function F(x), when differentiated, gives f(x). The answer is a function of x + c (where c is an arbitrary constant)