In comparison, an open manifold is a manifold without boundary that has only non-compact components.
The only connected one-dimensional example is a circle. The torus and the Klein bottle are closed. A line is not closed because it is not compact. A closed disk is compact, but is not a closed manifold because it has a boundary.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.
Abuse of language
This section possibly contains original research. (December 2019)
Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions),thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.
The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.