<PERSON>
As an Istar, and therefore a Maia, his true nature is of a holy spirit, and his physical form is just an incarnation, an avatar.
Nonetheless, after his fight with the Balrog of Moria, where he managed to kill it, he spent all his vital energy and his physical body died as well.
It is stated that <PERSON> himself intervened (presumably in an exceptional way, since this doesn't seem to be the norm) and granted his spirit to "go back" to his body once again, because he was the only Istar that stood true to their original mission.
This new reincarnation was not exactly the same as before: he was previously known as <PERSON>, but when he was brought back to life he did in the form of <PERSON>: his true potential was now revealed.
The wizards, also called Istari, were originally spirits of the order of the Maiar, the followers of the Valar. These were sent by the Valar to help and assist the peoples of Middle-earth against <PERSON> as he gathered his forces during the Third Age.
[...]
Their spirits remained immortal, and they aged only very slowly. However, their bodies could be killed: <PERSON> did indeed die from his duel with the Balrog of Moria, and only through the intervention of <PERSON> himself was he restored to his body.
<PERSON> Gateway, Wizards
<PERSON> used his last measure of strength to slay the <PERSON>, throwing him down the mountainside in ruin. <PERSON>'s spirit then left his body, having sacrificed himself to save the Fellowship.
[...]
But <PERSON>'s spirit did not depart Middle-earth forever at this time. As the only one of the five Istari to stay true to his errand, <PERSON> was sent back to mortal lands by <PERSON>, and he became <PERSON> once again. Yet, as he was now the sole emissary of the Valar to Middle-earth, he was granted the power to "reveal" more of his inner Maiar strength. This naked power that lay within him was seldom used during the remainder of his time in Middle-earth, as his mission was essentially the same: to support and succour those who opposed <PERSON>. 381
Nevertheless, when <PERSON>'s wrath was kindled his "unveiled" strength was such that few of <PERSON>'s servants could withstand him.
<PERSON> Gateway, <PERSON>
<PERSON>
<PERSON> was an Elf, thus immortal by nature: meaning that he was invulnerable to the passing of age, but could still be killed by violent acts or by great grief.
He fought with a Balrog during the Fall of Gondolin, on the mountains that encircled the hidden city, and managed to kill it, but he perished too when the dying Balrog grabbed him down the mountain side with it (this sounds familiar...).
Normally when an Elf died, his spirit went to the Halls of Mandos where it awaited to be re-embodied and stay in Aman.
<PERSON> was granted the unique chance to go back to Middle-Earth after his reincarnation.
<PERSON>’s spirit passed to the Halls of Mandos, where he waited with the spirits of the other Noldor who had died during their war against Morgoth. But because of <PERSON>’s noble actions in life, his reluctance at the Exile, and his furthering of the purposes of the Valar by saving <PERSON> and <PERSON>, he was re-embodied after only a short time. He had redeemed himself, and was purged of any guilt. Not only did his sacrifice get him an early pardon, it earned him great powers, so that he was almost an equal to the <PERSON>.
Eventually, <PERSON> sent him across the sea to Middle-earth, possibly as early as Second Age 1200, but more likely in 1600 with the Blue Wizards. If the latter date, he arrived just after the One Ring had been forged, <PERSON> built, and <PERSON> dead or soon to be so. While the Blue Wizards were sent to the east, <PERSON>’s mission was to aid <PERSON> and <PERSON> in the struggle against <PERSON>. He played a prominent behind-the-scenes role in the war in Eriador and the other struggles of the Second Age and Third Age. His part, though great, was mostly overlooked by the histories, because his immense, angelic power was not usually displayed openly. 381