The umbra (Latin for "shadow") is the darkest part of the shadow. From within the umbra, the light source is completely occluded. In astronomy, an observer in the umbra is said to be in the shadows experiencing total eclipse.
The penumbra (from the Latin paenes "almost, nearly" and umbra "shadow") is the region in which only a portion of the occulting body is obscuring the light source. An observer in the penumbra experiences a partial eclipse.
An alternative definition is that the penumbra is the region where some or all of the light source is obscured (i.e., the umbra is a subset of the penumbra). For example, NASA's Navigation and Ancillary Information Facility defines that a body in the umbra is also in the penumbra.
The antumbra is the region from which the occulting body appears entirely contained within the disc of the light source. If an observer in the antumbra moves closer to the light source, the apparent size of the occulting body increases until it causes a full umbra. An observer in this region experiences an annular eclipse.
