There are several conflicts between Telemachus and the suitors, and among the suitors themselves. All the suitors want to marry Penelope. Of course, only one could marry Penelope, which also leads to conflicts among the suitors themselves. If Penelope marries of the suitors, he would become Telemachus' new father, which would mean he gave up on Odysseus' being alive. He doesn't want that. He wants to be loyal to his great father, Odysseus. At the same time, the Greeks put a high value on hospitality, which makes it rather hard for Telemachus to just kick the suitors out, as this would be a breach of Xenia, or hospitality.
