In this book, the last of the so-called "Telemachia," some more names and references are thrown the readers' way with little to no explanation. Below is my attempt at clearing up confusion. To avoid duplication, here are the glossaries for Book I, Book II, and Book III. As is often the case in Homer, there are also several names whose larger import is lost to history, if they ever had a greater import at all; many of these fall into the category of "X son of Y" where X is someone who may or may not be important and Y is essentially just a name. These can be glossed over lightly.
Helen: Variously known as Helen of Sparta or Helen of Troy, the most beautiful woman in the world. The face that literally launched a thousand ships. Daughter of Leda and Zeus, who had taken the form of a swan to seduce her.
son of Achilles: Though not named, this is Neoptolemus, a major figure in his own right in several of the non-Homeric stories that deal with the fall of Troy and its aftermath. His name, conveniently enough, means "new warrior." If you're curious, he's a major character in several of the plays based on the Homeric myths, especially the Philoctetes of Sophocles, which is an excellent piece.
Hermione: The only child of Helen and Menelaus. Married to Neoptolemus, later (depending on who you ask) married to her cousin Orestes. Her name is derived from Hermes.
Cyprus, Phoenicia, the Egyptians, Ethiopians, Sidonians, Erembians, Libya: Variously, islands (Cyprus), nation-states (Phoenicia) and races (Ethiopians) in, around, and near the Mediterranean. The Sidonians were in modern day Lebanon, as were the Phoencians. The exact identity of the Erembians is lost to history.
Artemis: Goddess the Moon and of the hunt. A virgin goddess, she was the twin sister of Apollo. Called Diana by the Romans, by that name her legacy has stretched into neo-paganism and the ethos of the warrior woman.
Polybus: One of several mentioned in The Odyssey, I only mention him to point out that this one is a king in Egypt and is not to be confused with others that are mentioned or appear over the course of the poem.
Memnon: An Ethiopian king who fought on the side of Troy. He was the son of Eos, the dawn whose rosy fingers are never far away in Homer.
Aphrodite: Goddess of love and beauty--Venus to the Romans. In some ways, the instigator of the Trojan War, as she bribed Paris, Price of Troy with the hand of the most beautiful woman in the world, in order to win a beauty contest. (Seriously.) Unfortunately, Helen, wife of Menelaus, was acknowledged to be the most beautiful woman in the world, and Paris' abduction of Helen back to Troy to be his wife was the triggering event of the Trojan War.
Deiphobus: Prince of Troy, one of old King Priam's 50 sons. After the death of Paris but before the final fall of Troy, Helen was given to Deiphobus as wife for his deeds in the war. Helen did not love him as she had Paris. Accounts vary, but she either helped the invading Greek forces kill him or killed him herself.
Pharos: Island off the coast of Greece that would eventually be home to the famous lighthouse that bore its name, one of the Seven Wonders of the Ancient World.
Old Man of the Sea: In this instance, Proteus. (The term is used variously to refer to several water deities.) His name, with its connection to the prefix proto-, etc. strongly implies that he was a deity older than the Olympians, perhaps a Titan. His ability to change his shape gave English the word "protean." Wrestling with Proteus, or similar prophetic shape-shifting characters, recurs in several other Greek myths and is in fact part of a broader mythos that appears in mythological systems throughout the world.
Hera: Wife/sister of Zeus, and queen of the Olympian gods. Goddess of women and marriage--her Roman name was Juno. She was determinedly in the anti-Troy camp during the war, as she was one of the goddesses slighted by Paris when he named Aphrodite the most beautiful.
Elysian Fields: Also called Elysium. The land of the blessed dead, more particularly here, the "immortal" sons of Zeus (even immortals either went to Hades or were, in rare cases, take up to Olympus). Hades, the land of the dead, was, in later mythology, subdivided into several other areas to which spirits were assigned based on what sort of life they had led. Elysium was the best of those areas and the highest form of afterlife most mortals could hope for.
Rhadamanthys: Son of Zeus and Europa, brother of King Minos, ruler of the Elysian Fields in Hades' stead. In later mythology, one of the three judges of shades entering the underworld.
River Ocean: The ocean, particularly the Atlantic, thought by the Ancients (in Homer, at least) to be a vast river which encircled the known world.
Hephaestus: God of the forge and later, the volcano. (Roman name Vulcan). A lame craftsman god who was wedded to Aphrodite.
Medon: Herald and servant of Odysseus and Penelope.
Arcesius: Father of Laertes, grandfather of Odysseus, and former king of Ithaca. Notably, Zeus made his line one of "only sons," where Laertes was his only son, Odysseus his, etc., etc.