Argos was loyal to Odysseus by waiting twenty years for Odysseus to return, even though most had presumed he was dead. He was the first to recognize Odysseus when he returned from the Trojan War, even though Odysseus was disguised as a beggar. Argos would only die after he had seen his master once more.
Here is an excerpt about Argos' loyalty:
"Eumaeus, what a noble hound that is over yonder on the manure heap: his build is splendid; is he as fine a fellow as he looks, or is he only one of those dogs that come begging about a table, and are kept merely for show?"
"This hound," answered Eumaeus, "belonged to him who has died in a far country. If he were what he was when Odysseus left for Troy, he would soon show you what he could do. There was not a wild beast in the forest that could get away from him when he was once on its tracks. But now he has fallen on evil times, for his master is dead and gone, and the women take no care of him. Servants never do their work when their master's hand is no longer over them, for Zeus takes half the goodness out of a man when he makes a slave of him."