Select a region, choose Edit > Cut Image or Edit > Copy Image, then Edit > Paste Image.
See Create a mask by selecting a region.
You can also select a region of an image layer by creating a mask for that image layer (see Create a mask from an image file, Create a mask by painting, and Create a mask from curves).
To paste the cut/copied region (in Alias)
The cut/copied region is pasted into a new image layer above the active image layer.