Short answers, pulled from the story.
The Old Forest lies just beyond the Shire's eastern border near the centre of Eriador. It once formed the northern edge of one immense forest that reached all the way to Fangorn, hundreds of miles to the south-east.
Old Man Willow is a malign, sentient tree-spirit that grows beside the River Withywindle and holds influence over most of the Old Forest. He can hypnotize travelers and move his own roots and trunk, though he cannot walk from place to place.
The Old Forest first appeared in print in 1954, in the chapter titled "The Old Forest" from The Fellowship of the Ring.
The Bucklanders built the High Hay, a great hedge along Buckland's eastern border, because the Old Forest's trees had turned hostile toward them. They also cleared and burned a strip of trees outside the Hedge, which became known as the Bonfire Glade.
The Old Forest covers about 1,000 square miles, roughly 2,600 square kilometres. It is bordered by the Barrow-downs to the east, the Great East Road to the north, and the Brandywine river to the west and south.
Frodo Baggins entered the Old Forest on the advice of Merry Brandybuck, who had visited several times and held a key to its gate. Frodo used the forest to evade the pursuing Black Riders.
Peter Jackson's film adaptation cut the Old Forest entirely, along with the related Crickhollow chapter, the Barrow-downs, and the character Tom Bombadil.