A periodic curve is basically a curve that has the first d and last d control points (where d is the degree of the curve) the same, together with a non-clamped knot vector. In Rhino though, you see the duplicate control points as single control points, which makes it easier for the user to manipulate periodic curves.
This in contrast to closed and open curves, that have a clamped knot vector. This means that the first and last control point are also the start and end of the curve, respectively. A closed curve has only a two overlapping control points (first and last).