What is the significance of lines 1166-1168 "trained as a puppy by Odysseus, but never taken on a hunt before his master sailed for Troy?"
The quotation in this question comes from Book 17 of Homer’s Odyssey. In this book, Odysseus has returned to his native island of Ithaca. Unfortunately, his house is infested with the suitors of his wife Penelope, so Odysseus goes back to his home disguised as a beggar. When he reaches his house, he sees an old dog on heaps of dung and covered with ticks. This dog, Argus, was a puppy when Odysseus left for Troy. At some point in his life, Argus lived up to the meaning of his name ("swift"). Eumaeus tells the diguised Odysseus:
"If he had the form and vigour he had when Odysseus left for Troy you’d be amazed by the speed and power. He was keen-scented on the trail, and no creature he started in the depths of the densest wood escaped him" (A.S. Kline translation).
Alas, this aged hound is now on the verge of death. Argus, however, seems to sense Odysseus' presence. Accordingly, the dog's tail begins to wag and his ears move. Odysseus recognizes the dog and wipes a tear from his eye. Odysseus must maintain his disguise and cannot respond to Argus in the way that a master would like to acknowledge his dog. After the disguised Odysseus enters the palace, Argus' life comes to an end:
"As for Argus, seeing Odysseus again in this twentieth year, the hand of dark death seized him" (A.S. Kline translation).