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Achilles is playing a game with Ajax. Presumably, it's a gambling game. They're armed, though, and ready for battle. The photographer notes that this was a popular theme of the late 500s B.C.
Achilles and Ajax were both major heroes of the Greeks during the Trojan War. Both die during the war, Achilles by a divinely-guided arrow shot by the Trojan prince Paris into his Achilles' heel, and Ajax dies by suicide when he was driven mad by Athena to prevent the warrior from killing his fellow Greeks.
The madness came after the decision to award the armor of the late Achilles to Odysseus, instead of Ajax, who wanted it and felt he had earned it.
Genealogy of Achilles
For more on the genealogy of Achilles, see Achilles Family Tree. Among other notables on the tree, Tantalus may have been the great-great-great grandfather of Achilles, through his son Pelops, since Pelops was, perhaps, the father of Sciron. However, Sciron is best known for coming to the attention of the crime-fighter Theseus*. Another genealogy puts Chiron in Sciron's place, so when Achilles is fostered out to the centaur, Achilles is being kept within the extended family.
[E.1.2] Fourth, he slew Sciron, the Corinthian, son of Pelops, or, as some say, of Poseidon. He in the Megarian territory held the rocks called after him Scironian, and compelled passers-by to wash his feet, and in the act of washing he kicked them into the deep to be the prey of a huge turtle.
[E.1.3] But Theseus seized him by the feet and threw him into the sea.
Apollodorus Epitome
Relationship Between Achilles and Patroclus
Peleus' grandmother, Aegina, is the ancestor of Achilles' friend Patroclus. By some accounts, Patroclus is the son of Menoetius, son of Actor and Aegina. This makes Peleus, who is the son of Aeacus, son of Zeus and Aegina, and Patroclus half cousins, and Achilles and Patroclus half cousins once removed.
As for most of Greek mythology, Timothy Gantz is a great source. According to Gantz, Pindar makes Aegina the mother of Aeacus and fragments of the Hesiodic corpus make Aeacus grandfather of Patroclus.
Peleus and Thetis - Parents of Achilles
Thetis was a sea nymph, specifically, a Nereid who inherited the ability to shape-shift. She helped (1) Hephaestus, when he was thrown from Olympus, (2) Zeus, when threatened by other gods, and (3) Dionysus, when he fled from Lycurgus. Poseidon and Zeus were both interested in Thetis until a prophecy revealed that a son born to her would be greater than the father. So instead of mating with the gods, Thetis was pressed to marry the Thessalian King Peleus.
Thetis appears not to have been too happy with the arrangement and when Peleus came to take her away, she changed her shape, again and again. In time, she agreed to marry Peleus.
Another story has Thetis reject Zeus' offers out of loyalty to Hera. Arranging Thetis' marriage to Peleus was Zeus' revenge.
The son of the union of Peleus and Thetis was the greatest Greek hero of his generation, Achilles.
Achilles Kills Memnon
Memnon was an Ethiopian king on the Trojan side in the Trojan War. Achilles killed him in revenge (as Achilles also did with Hector after Patroclus was slain), after Memnon killed Nestor's son Antilochus. Memnon had refused to fight Nestor when challenged by the aggrieved father because the king of Messenia was quite old. Achilles stood in for him, although he had been warned his own death would soon follow Memnon's.
Memnon was the son of the Titan goddess of the dawn, Eos.
Achilles and Patroclus
Achilles and Patroclus were close friends from their time being fostered by Chiron. They were also cousins of some sort and possibly lovers.
Agamemnon had angered Achilles, so Achilles was sitting out the Trojan War, but Patroclus tried to talk him into rejoining or, if not, at least to lend him his armor and let him lead the Myrmidons into battle. Achilles agreed to let Patroclus fight dressed in his armor and to lead the Myrmidons.
Patroclus went into battle looking just like Achilles, at least to the Trojans. The Trojans were afraid of Achilles because he was the greatest of the Greeks. Having him sitting out the war was good for the Trojans. Having him back fighting was dangerous. That made the Achilles figure who was secretly Patroclus a prized Trojan target. Although Patroclus wasn't as good a warrior as Achilles, he did kill Sarpedon and many other Trojans.
Patroclus was killed, ultimately, by Hector.
After Achilles got revenge of his friend's killing by killing Hector, he cremated Patroclus' corpse and held elaborate funeral games to honor him.
Thetis Brings Armor to Achilles
When Patroclus was killed wearing Achilles' armor, Achilles needed a new set. Thetis went to the blacksmith god Hephaestus, who owed her a favor, to ask him to make Achilles a phenomenal set. It's the divinely-forged armor that Achilles' nymph-mother Thetis brings her son.
Achilles is clearly saddened by the death of his friend in this picture.
Achilles Kills Hector
Achilles sent his beloved Patroclus into the fray dressed in his armor. The Trojans saw Achilles' insignia and assumed Patroclus was Achilles, which made him a focal point. Not being anywhere near the warrior that Achilles was, Patroclus died, killed in part by the leading warrior of the Trojans, the heir-apparent, Prince Hector.
Achilles' reaction was rage mixed with profound grief, but it was enough to shake him out of his indifference and rejoin the battle.
He fought one-on-one against Hector until Hector died. Then Achilles attached him to his chariot and dragged him through the sand and dirt until he had reduced his rage. King Priam, father of Hector, went to Achilles to beg for the return of the mangled corpse of his son. Achilles was persuaded to do so in order that Hector might receive proper burial; however, as far as the mangling goes, the gods had prevented Achilles' actions from being effective. They had kept the corpse of Hector intact.
In the mosiac, Achilles' mother Thetis is about to give her infant a bath. The AX appears over the ruined area of the mosaic, but stands for Achilles, who seems to be further to the left on a lap.
Thetis was a nymph whom both Zeus and Poseidon wished to marry, but a prophecy revealed that the son of Thetis would be greater than the father, so both Poseidon and Zeus stepped down in favor of a noble human, King Peleus.
Thetis was awarded Peleus by Zeus for noble behavior, but Thetis was displeased with having to marry a mortal. Artistic depictions of the wooing show Peleus clinging to a shapeshifter. Peleus proves up to the challenge and they wed. The marriage of Thetis and Peleus was a grand affair on Mt. Pelion, with all the gods and goddesses. Unfortunately, the list of guests had one important omission, Eris, goddess of discord. In response to the slight, she gave a gift of a golden apple to the most beautiful of the goddesses. This led to the Judgment of Paris, abduction of Helen, and the Trojan War.
As to the maternal behavior of Thetis... after her attempts to immortalize her infant, as per this bath, dipping him in the River Styx, or burning off his mortality, were interrupted, Thetis took off in a huff*, leaving Achilles in the care of his father.
Peleus took the instructional course most popular for young heroes. He farmed him out to the centaur Chiron for fostering.
*In some accounts, Thetis and Peleus live together during Achilles' upbringing.
Thus, Thetis is there to see Achilles off to war.
Achilles dies during the Trojan War (but after the action of the Iliad) mortally wounded by an arrow shot by Paris. Ovid (Metamorphoses 12) has Apollo urge Paris to shoot at Achilles and then guide his aim. Other writers allow Paris to do the shooting (or stabbing) alone, or Apollo, or Apollo disguised as Paris. Apollodorus and others say the wound was in Achilles' heel. Not all the authors subscribed to the idea that Achilles was only mortal in his heel, especially since it doesn't make a lot of sense to think that an ordinary wound in the ankle would be lethal.
The bronze man Talos, however, did die when the nail in his ankle was removed and all the life-giving fluid running through his body leaked out. That Achilles' mother was a nymph made Achilles a demi-god, at best. Her attempts to make him immortal by burning or immersion in the River Styx were obviously not entirely successful.
Frazer's notes to Apollodorus go through the variants and authors.
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