The suitors in the Odyssey are men who want to marry the “widowed” Penelope. They never succeed. In the end, they are killed by Odysseus for squandering his wealth and harassing Penelope.
Detailed answer:
Odysseus has been away from his palace for the last twenty years. Still, his wife Penelope does not lose hope that she will reunite with her long-gone husband one day. It is almost like she is the only one believing in his return. Others consider her a “widow” and, hence, available for marriage. She soon has hundreds of suitors asking for her hand, but she rejects them all. The reader learns the names of only a few of them. Some of them are decent, like Amphinomus, and others, such as Antinous, have evil intentions.
Overall, the suitors in the Odyssey are far from gentle, romantic types. Homer depicts them as ill-mannered, piggish men. Their presence is not flattering but oppressive, and Penelope and Telemachus have to unite against them. The suitors serve no useful purpose in the palace. They are running amok through the palace, eating the food, slaughtering the livestock, and harassing the widow.
Indeed, Penelope cannot chase them away from her house – that would be rude. Instead, she comes up with smart “ruses” or tricks to postpone her decision. First, she tells them to wait until she finishes the funeral shroud for Odysseus’ father. However, after three years, her lie becomes uncovered. So instead, she organizes an archery contest. She promises to become the wife of whoever can string Odysseus’s bow through 12 axes. None of the suitors can do that, and the only one who achieves it turns out to be Odysseus. He was disguised as a beggar to keep a low profile. Penelope finally recognizes him and happily reunites with her husband.