a manifold with boundary
An ordinary manifold looks locally like all of R^n — every point has a little patch indistinguishable from open space, with no edge. A manifold with boundary allows some points to sit on an edge: near such a point the space looks like a closed half-space, R^n with one coordinate required to be >= 0. The closed disk, a solid ball, and a cylinder of finite length are the standard examples; their rims, spheres, and end-circles are the boundary.
Precisely, an n-manifold with boundary M is modeled on the half-space H^n = {(x^1, ..., x^n) : x^n >= 0}: every point has a neighborhood diffeomorphic either to an open subset of R^n (interior points) or to an open subset of H^n meeting the hyperplane x^n = 0 (boundary points). The boundary, written partial M, is the set of boundary points; it is itself a manifold WITHOUT boundary of dimension n - 1, and partial(partial M) is empty. An orientation of M induces a canonical orientation on partial M via the 'outward normal first' convention: an oriented basis of partial M is positive when preceded by an outward-pointing vector it makes an oriented basis of M.
Manifolds with boundary are the natural domains for Stokes' theorem, which relates an integral over M to an integral over partial M — so getting the induced boundary orientation right is exactly what makes Green's, Gauss's, and the classical Stokes theorems come out with correct signs. A point of care: the boundary partial M is NOT the topological boundary of M sitting inside some ambient space; it is intrinsic, defined by the local half-space model. Also, the interior of M (points modeled on open R^n) is an ordinary manifold, and partial M can be empty, in which case M is just an ordinary closed manifold.
The closed unit disk D^2 = {(x, y) : x^2 + y^2 <= 1} is a 2-manifold with boundary; its boundary partial D^2 is the unit circle S^1, a 1-manifold without boundary. The solid ball B^3 has boundary the sphere S^2. The half-open cylinder S^1 x [0, 1] has boundary two disjoint circles S^1 x {0} and S^1 x {1}. In each case the induced orientation on the boundary, via outward-normal-first, is exactly the one that makes Stokes' theorem hold sign-correctly.
The boundary of an n-manifold-with-boundary is an (n-1)-manifold without boundary; partial(partial M) is always empty.
partial(partial M) = empty is the geometric mirror of d^2 = 0; the two facts are dual under Stokes. The boundary is an intrinsic notion from the half-space model, not the topological frontier in an ambient space — an open interval (0,1) has no manifold boundary even though as a subset of R it has endpoints.