"It is a shroud for Lord Laertes"
Penelope is weaving a funeral shroud for Odysseus' father, Laertes. This was a vital role for a Greek woman, who was expected not only to take care of domestic duties in the household, but also to participate in mourning rituals. Burial, with proper funeral rites and mourning, was incredibly important to the Greeks. For a family to shirk these responsibilities would have been considered the height of shameful behaviour.
Click here for a presentation on Greek women weaving and spinning from the British Museum.