How do I acquire an immortal horse?

Shadowmere is a horse in The Elder Scrolls V: Skyrim, originally owned by Lucien Lachance. Astrid lets the Dragonborn borrow Shadowmere to travel to the Dawnstar Sanctuary for the quest The Cure for Madness. She then gives him to the Dragonborn as a reward for completing the quest

From http://elderscrolls.wikia.com/wiki/Shadowmere_%28Skyrim%29

Regarding Shadowmere's immortality:

Shadowmere is not essential and can die, unlike his counterpart in Oblivion. However, 10 days later, he will respawn at the location of his death. If Shadowmere dies in a river or lake, he will respawn after 10 days in the black pool near the Dark Brotherhood Sanctuary.

Source

If you have the Dawnguard DLC you can obtain Arvak, an undead horse.

Arvak is a horse that can be summoned from the Soul Cairn after completing Soul Cairn Horse Quest. He can be summoned with the apprentice level conjuration spell Summon Arvak.

All you need to do to obtain him is progress through the Dawnguard main quest until you are able to enter the Soul Cairn, then complete This quest.