"As a third labour he [Eurystheus] ordered him to bring the Cerynitian hind alive to Mycenae. Now the hind was at Oenoe; it had golden horns and was sacred to Artemis; so wishing neither to kill nor wound it, Hercules hunted it a whole year. But when, weary with the chase, the beast took refuge on the mountain called Artemisius, and thence passed to the river Ladon, Hercules shot it just as it was about to cross the stream, and catching it put it on his shoulders and hastened through Arcadia. But Artemis with Apollo met him, and would have wrested the hind from him, and rebuked him for attempting to kill her sacred animal.Howbeit, by pleading necessity and laying the blame on Eurystheus, he appeased the anger of the goddess and carried the beast alive to Mycenae." [2.5.3]
It seems that Euryteus (and Hera, in some versions) were becoming aware that the hope they carried that Hēraklēs would perish at the hands (or teeth) of a monster became a vain hope. If not even the Hydra could end his life, very few other thing would have stood a chance. Changing tactics, Eurystheus gave Hēraklēs a different kind of task: one in which the task itself was not dangerous, and only required a sharp mind. Hēraklēs wasn't a fool: he was well aware that capturing an animal sacred to Artemis was as deadly a task as his previous ones. In fact, it might be more lethal: upsetting the Theia of the hunt had gotten far more powerful men killed.
Unable to refuse, Hēraklēs went off in search of the golden hind of Artemis, which was also known as the Keryneian Hind (Κερυνῖτις ἔλαφος), or Krynites. In his 'Hymn to Artemis', Kallimachos describes the sacred animal, and explains that there were more of these animals--five in total--of which Artemis captured four and yoked them to her chariot. The other escaped:
"Thence departing (and thy hounds sped with thee) thou [Artemis] dist find by the base of the Parrhasian hill deer gamboling – a mighty herd. They always herded by the banks of the black-pebbled Anaurus – larger than bulls, and from their horns shone gold. And thou wert suddenly amazed and sadist to thine own heart: “This would be a first capture worthy of Artemis.” Five were there in all; and four thou didst take by speed of foot – without the chase of dogs – to draw thy swift car. But one escaped over the river Celadon, by devising of Hera, that it might be in the after days a labour for Heracles, and the Ceryneian hill received her." 
Hēraklēs found a worthy adversary in the hind, which eluded him for almost a year. He chased it to the far corners of Hellas and beyond, and then, eventually, he came upon it. How, exactly, the hind was caught varies, and even varied in ancient times. Diodorus describes:
"The next Command which Heracles received was the bringing back of the hart which had golden horns and excelled in swiftness of foot. In the performance of this Labour his sagacity stood him in not less stead than his strength of body. For some say that he captured it by the use of nets, others that he tracked it down and mastered it while it was asleep, and some that he wore it out by running it down. One thing is certain, that he accomplished this Labour by his sagacity of mind, without the use of force and without running any perils."
Whatever the case, the hind is caught, and Hēraklēs hurries to the king to finish the task. On the way back, however, Artemis and Apollon appear, both rady to end his life. Hēraklēs pleads with Them, lying the blame on Eurystheus. Artemis makes him promise to return the hind to Her when he has shown it to the king, and Hēraklēs agrees. The divine twins spare his life, and thus foil Eurystheus' plan to have Hēraklēs killed.
So Hēraklēs returns with the stag on his shoulders. The king is furious, but he is happy he will at least possess the golden hind now. He orders Hēraklēs to put the hind in the stables, but Hēraklēs knows that the hind must be returned to Artemis without delay. Devising a plan of his own, he tells the king to come and take the hind from him. When he releases the animal, he does so slightly too early, and the swift-footed hind escapes into the woods. Hēraklēs has completed another labour, and he has managed to appease Artemis; not bad for someone previously known mostly for his brute strength.
Image source: sculpture