The reason that Owl is named Wol is because Wol is a Kentish and Sussex dialect word for Owl, which Milne would have been familiar with, living on a farm at Hartfield at the time he was writing Winnie the Pooh.
Owl is a good friend of Winnie the Pooh, Christopher Robin, and all the other inhabitants of the Forest. He is always happy to offer his opinions, advice, and anecdotes - whether or not they are actually wanted. Owl also enjoys telling stories about his relatives, including his aunt who laid a seagull's egg by mistake and his Uncle Robert who once survived a very blusterous day.
In the Winnie-the-Pooh book, Owl lives in a tree known as The Chestnuts, located in the middle of the Hundred Acre Wood and described as an "old world residence of great charm" which is grand enough to have both a door-knocker and a bell-pull. That house is blown down by a storm in the eighth chapter of The House at Pooh Corner. Eeyore eventually discovers what he believes is the perfect new house for Owl, apparently without noticing that it is actually Piglet's house. Nonetheless, Piglet offers the house to Owl, and he presumably moves in. Owl made a sign indicating that he planned to call his new house "The Wolery".
Unlike most of the original cast of the books, the illustrations of Owl look more like a living animal and than a stuffed one. This idea is also supported by Rabbit's comment to him, "You and I have brains. The others have fluff." In Ernest H. Shepard's illustrations, Owl appears to be about a head shorter than Pooh, and a little below hip-height to Christopher Robin. He is sometimes but not always drawn wearing reading glasses. When the illustrations show him writing, he holds the pen in his talons, not with his wing.
Owl appears in chapters IV, VI, VIII, IX, and X of Winnie-the-Pooh. He also appears in chapters V, VIII, IX, and X of The House at Pooh Corner, and is mentioned in several other chapters.
Owl has never been seen on My Friends Tigger & Pooh.