What is the name of the horse given to Gandalf by the Rohirrim?

The horse given to Gandalf by the Rohirrim is Shadowfax.

Shadowfax is one of the Mearas, the exceptionally swift and intelligent horses of Rohan. After Gandalf escapes Saruman and reaches Edoras, King Théoden allows him to choose a mount. Gandalf selects Shadowfax, who becomes his loyal companion and carries him across Middle-earth.

Shadowfax is not merely a fast warhorse. Tolkien describes him as a great silver-grey stallion who understands human speech and will not ordinarily accept a rider, saddle, or bridle. He carries Gandalf by choice, making their partnership unusual even among the Rohirrim's famed horses.

A common mix-up is with Snowmane, Théoden's horse, or Windfola, Éowyn's mount. Shadowfax later carries Gandalf during the War of the Ring, including his arrival at Minas Tirith, and is present when Gandalf departs Middle-earth from the Grey Havens.