Short answers, pulled from the story.
Týr is a god in Germanic mythology and a member of the Æsir, the principal group of Norse deities. He is described in the Prose Edda as the bravest and most valiant of the Æsir, associated with victory in battle, justice, and the Germanic thing assembly. His name derives from the Proto-Germanic Tīwaz, meaning simply "God."
Tuesday comes from the Old English tīwesdæg, meaning "day of Tīw," which is a translation of the Latin Martis dies ("Day of Mars"). Early Germanic peoples identified their god Tīwaz with the Roman war god Mars and adopted the Roman seven-day week, substituting Tīwaz's name for Mars in the Tuesday slot.
Týr lost his right hand when the gods sought to bind the monstrous wolf Fenrir with the magical fetter Gleipnir. Fenrir agreed to be bound only if one of the gods placed a hand in his mouth as a pledge. Týr volunteered, and when the gods refused to release Fenrir, the wolf bit off Týr's hand at the wrist, which the Prose Edda calls "the wolf-joint."
The Tiwaz rune represents the sound /t/ in the runic alphabets used by ancient Germanic peoples before they adopted the Latin alphabet. Týr is the namesake of this rune, and on runic inscriptions the character functions frequently as a magical symbol. The name appears earliest as tyz in the 4th century Gothic alphabet.
Roman writers referred to Týr as Mars through the process of interpretatio romana, identifying the Germanic deity with the Roman war god. A 3rd century Latin inscription at Housesteads Roman Fort on Hadrian's Wall names the deity as Mars Thingsus, with the epithet Thingsus reflecting Týr's association with the Germanic thing, a traditional legislative assembly.
At Ragnarök, the dog Garm, described in the Prose Edda as the most evil creature, breaks free from its binding at Gnipahellir. Týr and Garm fight, and the two are foretold to kill each other.