As James Joyce is a master of incorportating subtle allusions into his writing (as is evidenced by the 500+ page reference work Allusions in Ulysses by Weldon Thornton), perhaps the most important allusion connected to this novel is its title: Ulysses. Obviously, this title is a reference to Homer's Odyssey, whose main character, Odysseus (or Ulysses), partakes in an epic quest to make his way home to Ithaca to be reuinted with his wife, Penelope, and his son, Telemachus. In each chapter of the novel, Joyce subtly incorporates references to Homer's Odyssey which many readers miss without the help of texts such as Daniel Schwarz's Reading Joyce's Ulysses.
Foremost, Joyce constructed his characters upon those in Homer's Odyssey. Stephen Dedalus, the novel's protagonist, is a pensive, intellectual young man whose preoccupation with his mother's recent death haunts him, and whose relative estrangement from his own father leaves a void in his life. Stephen's role as a somewhat "lost" son allows readers to identify him with Telemachus, Odysseus's son who fends off his mother's suitors in his father's absence.
Leopold Bloom represents the Ulysses figure, as he is a husband (although his wife, Molly--who represents Penelope-- is having an affair with Blazes Boylan and Leopold is carrying on a somewhat inappropriate relationship with another woman) and a father. He has a daughter, Milly, with whom he has difficulty identifying, and he lost a son, Rudy, in infancy. Thus, it can be said that Bloom is in search of a son and reconciliation with his wife, just as Odysseus is determined to make his way back to Telemachus and Penelope.
Joyce connects each episode with some aspect of Homer's Odyssey. For example, Chapter 6 alludes to Homer's Hades episode, as Bloom attends the funeral of a friend and reflects on the feelings and customs associated with death, and specifically thinks of the death of his infant son and the suicide of his own father. Similarly, the final chapter in the novel, which is comprised of only 8 sentences in stream-of-consciousness format, marks Molly's acceptance of Bloom back into her bed, just as Penelope welcomes Odysseus home after years of his attempts to return. Again, these allusions are very subtle, and readers studying the book need to read texts like Schwarz's in order to understand the varying layers of Joyce's text.
For more information on each chapter's title and allusions to Homer's Odyssey, see the enotes link below. Other, more detailed reference texts will be necessary if you truly want to study and understand the novel. Ultimately, though, Joyce is too complex a writer to simply tell readers what they should be getting out of his novels--particularly Ulysses and Finnegans Wake. However, one of the most rewarding experiences for a reader of Ulysses involves understanding the complexity in Joyce's construction and appreciating the seemingly effortless way he's able to interweave such a vast allusion so artfully.
This is the breakdown as I understand it:
Part I: The Telemachiad
2.1.1 Episode 1, Telemachus
2.1.2 Episode 2, Nestor
2.1.3 Episode 3, Proteus
Part II: The Odyssey
2.2.1 Episode 4, Calypso
2.2.2 Episode 5, The Lotus Eaters
2.2.3 Episode 6, Hades
2.2.4 Episode 7, Aeolus
2.2.5 Episode 8, The Laestrygonians
2.2.6 Episode 9, Scylla and Charybdis
2.2.7 Episode 10, The Wandering Rocks
2.2.8 Episode 11, The Sirens
2.2.9 Episode 12, The Cyclops
2.2.10 Episode 13, Nausicaä
2.2.11 Episode 14, The Oxen of the Sun
2.2.12 Episode 15, Circe
Part III: The Nostos
2.3.1 Episode 16, Eumaeus
2.3.2 Episode 17, Ithaca
2.3.3 Episode 18, Penelope