The primary strength of Tolkien (for me and others I am sure) have always been the variety of characters. Some obvious, some probably not so. Mostly taken from a writer-reader perspective.
Gandalf. You would think it is easy to make a likable old wizard like him, but there are few (if any) other writers who achieved that.
Beorn. Very special, the way of life represented nicely there.
Boromir. Love the dilemma, love the humanity in the failure, love the redemption.
Sméagol/Gollum. The hate and pity aspect is perfect.
Túrin Turambar. Great and tragic, there are similarities to Aragorn's story, the latter being much more lucky, the former much more powerful.
Fingolfin. Sheer power gazing from almost every line he's in. How I wish Tolkien actually had finished the Silmarillion himself.
Glorfindel. Sort of a gateway to the great old days.
Gildor Inglorion. Again, just a few lines in the story and a very memorable character springs up.
Bilbo Baggins. The transition is close to unbelievable but just believable enough.
Treebeard. Ents rock (and stone).
Huan. Another strength, why not put in a real bad-arse dog, kicking Sauron's butt in the process, and of course dying emotionally as is the way with dogs.
Tom Bombadil. The best enigma there ever was.