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When Snape learned that Lily Potter was in danger, he approached his enemy Dumbledore (remember, Snape was a Death Eater back then). In exchange for protecting Lily, Dumbledore asked for the loyalty of Snape.
But then, Dumbledore failed to protect Lily. It doesn't matter if it was Dumbledore's mistake or not, Lily was dead.
Why would Snape then give his loyalty to Dumbledore?
No one was expecting Pettigrew's betrayal. Remember that it was Lily and James who made Peter the Keeper a few days before the attack. Dumbledore likely didn't know about this, and even if he did, it seemed a reasonable choice at the moment.
As to why he continued to work for Dumbledore, this is the moment when Dumbledore asks Snape to help him protect Harry:
“And what use would that be to anyone?” said Dumbledore coldly. “If you loved Lily Evans, if you truly loved her, then your way forward is clear.”
Snape seemed to peer through a haze of pain, and Dumbledore’s words appeared to take a long time to reach him.
“What — what do you mean?”
“You know how and why she died. Make sure it was not in vain. Help me protect Lily’s son.”
“He does not need protection. The Dark Lord has gone —”
“The Dark Lord will return, and Harry Potter will be in terrible danger when he does.”
There was a long pause, and slowly Snape regained control of himself, mastered his own breathing. At last he said, “Very well. Very well. But never — never tell, Dumbledore! This must be between us! Swear it! I cannot bear . . . especially Potter’s son . . . I want your word!”
Harry Potter and the Deathly Hallows, chapter 33 "The Prince's Tale"; emphasis mine.
So Dumbledore offered Snape to preserve Lily's legacy, as this is what Snape saw Harry as - Lily's child. Not James', but Lily's; a way to honour her, and a way for him to remember her, by keeping a piece of her nearby. Remember this iconic tearjerker scene:
“But this is touching, Severus,” said Dumbledore seriously. “Have you grown to care for the boy, after all?”
“For him?” shouted Snape. “Expecto Patronum!”
From the tip of his wand burst the silver doe: She landed on the office floor, bounded once across the office, and soared out of the window. Dumbledore watched her fly away, and as her silvery glow faded he turned back to Snape, and his eyes were full of tears.
“After all this time?”
“Always,” said Snape.
Ibid; emphasis mine.
In the films, Dumbledore additionally says "If you truly loved her..." which seems to swing Snape to his side (starts at 2:43):
I think something that is not expressed in Gallifreyan's excellent answer has to do with Snape's relationship with Voldemort.
Snape's loyalty to Voldemort was never as deep as his loyalty to Lily. Since Snape was always primarily on Lily's side, Voldemort's ambitions to harm Lily drew a line in the sand that Snape could not cross. He would certainly never return his loyalty to the one that murdered Lily. Dumbledore knew this and therefore knew he could always trust Snape so long as Dumbledore himself remained on the side of Lily and company.
People don't always keep promises purely out of obligation. He could have not helped as Dumbledore had failed, but it was never a 'contract'. If Snape had refused to help, Dumbledore wouldn't have let Lilly die. Or out of revenge, after Lilly died they both wanted to do the same thing, kill Voldemort.
Because until this point Snape saw Voldemort as power - he didn't understand the difference between Good and Evil (or specifically didn't understand the 'Magic' of Love). Such a powerful experience as losing Lily, whom he loved deeply, allowed Snape to feel proper remorse.
Through proper remorse and the power of love, Snape's loyalties changed from the (what he now recognized as) flawed ways of Voldemort, to the selfless side of 'love'; Dumbledore.
