For example, let F and K be finite fields. The extension F/K is galois, and the galois group is cyclic.
A field extension is solvable if it is finite, and galois, and its galois group is solvable.
Every cyclotomic extension has an abelian galois group, and is solvable. In many cases the extension is cyclic, e.g. adjoining the pth root of 1 where p is prime.
Intuitively, a solvable extension is a tower of cyclic extensions. Let's look at this more closely.
Let F/K be galois with galois group G, and let H be a normal subgroup of G, having fixed field L between F and K. We can now examine F/L, with galois group H, and L/K, with galois group G/H. If G is finite, we can create a tower of intermediate field extensions, according to its composition series. Each extension is galois, with a simple galois group. Therefore an extension F/K is solvable iff it is a tower of intermediate cyclic extensions, each having prime order.