Short answers, pulled from the story.
Old Man Willow is a malign tree-spirit of great age in Tom Bombadil's Old Forest, appearing physically as a large willow tree beside the River Withywindle. He is the first hostile character the Hobbits encounter after leaving the Shire. His grey thirsty spirit spreads dominion over nearly all the trees from the Hedge to the Barrow-downs.
Old Man Willow first appeared in Tolkien's poem "The Adventures of Tom Bombadil", published in 1934 in The Oxford Magazine. In an early 1938 draft of The Lord of the Rings, the tree and spirit were still two separate entities; they fused into a single indivisible being in later drafts.
Peter Jackson explained in From Book to Script that Old Man Willow was cut because the character was not advancing the story of Frodo carrying the Ring. Ralph Bakshi's 1978 animated film had already omitted the Old Forest entirely, setting a precedent Jackson followed.
Old Man Willow casts a spell of drowsiness over the Hobbits. He traps Merry and Pippin inside cracks in his trunk and tips Frodo into the River Withywindle. Tom Bombadil rescues them by singing to the tree until it releases Merry and Pippin.
Philologist Jason Fisher traces the name to Old English. Withy derives from wiþig, meaning a willow or flexible twigs woven into wicker baskets. Windle comes from Old English windel-treow, the willow or basket-maker's tree, and is a cognate of the modern verb to wind. Fisher reads Withywindle as perhaps meaning the willow-winding river.
Terence Tiller's 1955-1956 BBC radio play included Old Man Willow, though Tolkien disliked the production and no recording survives. The 1991 Soviet television play Khraniteli also included him, with Old Man Willow trapping two Hobbits in the Old Forest as Tolkien wrote.