Symbolism, Imagery, Allegory
Sometimes a bow is just a bow—and sometimes it's a symbol of kingship and virility, like when Penelope sets up a content so that the man who can string the bow and shoot an arrow through twelve axe heads will win her hand in marriage. Really, Homer practically hits us over the head with it:
First, Telemachos struggles but eventually "would have strung it" (21.128), when Odysseus stops him. (Good idea, if you ask us—that would be kind of disturbing.) Then, the suitors start failing, and Eurymachos spells it out, in case we still don't get it: "it is not so much the marriage I grieve for … it is the thought, if this is true, that we come so far short of godlike Odysseus in strength, so that we cannot even string his bow" (21.250-25). And what does Odysseus do? He strings that bow just like a man stringing a lyre, "without any strain" (21.409).
Well, we sure know who Penelope's going home with tonight.