Short answers, pulled from the story.
The most widely accepted meaning of Yggdrasil is "Odin's horse", interpreted as a kenning for gallows. The element Ygg is one of Odin's names, and drasill means "horse". Alternative scholarly interpretations include "tree of terror, gallows" (F. Detter) and "yew pillar" (F. R. Schröder).
According to Grímnismál and the Prose Edda, the three roots of Yggdrasil extend to three separate locations: one to the well Urðarbrunnr in the heavens, one to the spring Hvergelmir beneath Niflheim, and one to the well Mímisbrunnr, which is said to hold wisdom and intelligence.
Yggdrasil is home to the dragon Níðhöggr at its roots, the squirrel Ratatoskr running along its trunk, an unnamed eagle with the hawk Veðrfölnir between its eyes at the top, and four stags named Dáinn, Dvalinn, Duneyrr, and Duraþrór feeding on its branches. Multiple serpents are also listed in Grímnismál as gnawing at the tree below.
In Völuspá, the völva describes Yggdrasil shivering and groaning as Ragnarök begins. The scholar Rudolf Simek argues that two survivors, Líf and Lífþrasir, hide inside Yggdrasil during the catastrophe, identifying the hiding place called Hoddmímis holt as an alternate name for the world tree itself.
In stanza 138 of Hávamál, Odin describes hanging on a "windy tree" for nine nights, wounded by a spear, dedicated to himself, in order to seize the runes. Scholars near-universally identify this tree as Yggdrasil, and the name itself, meaning "Odin's gallows", directly references this event.
Yggdrasil is attested in the Poetic Edda, compiled in the 13th century, where it appears in the poems Völuspá, Hávamál, and Grímnismál. It also appears in the Prose Edda, compiled in the 13th century by Snorri Sturluson, primarily in the books Gylfaginning and Skáldskaparmál.