A monogenic number field is a number field whose ring of integers admits a power basis.
Such a basis is often called a power integral basis.
All quadratic fields and cyclotomic fields are monogenic. Further, the maximal real subfield of a cyclotomic field is monogenic.
An example, due to Dedekind, of a non-monogenic field is the field generated by a root of x^3-x^2-2x-8.
The problem of characterizing all monogenic number fields is known as Hasse's problem.
Below is a collection of some known families of monogenic fields. Unless otherwise mentioned the power basis is generated by a root of the polynomial given below.
Please send me an email if you have any corrections or families you'd like to see included.