There are two slightly different concepts of cyclotomic extensions: these can mean either extensions formed by adjoining roots of unity, or subextensions of such extensions. The cyclotomic fields are examples. Any cyclotomic extension (for either definition) is abelian.
If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker-Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.
There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.
