Strother Martin. I figured somebody would have mentioned him by now since Cool Hand Luke was mentioned in the OP.
He celebrated his birthday last year, and will be celebrating another in a couple of weeks. I’ve seen no indication he wasn’t still alive.
He was an incredibly busy actor in his time – he’s talked about two movies a day, each with a single scene and line. Always recognizable, too, with a raspy voice and sharp profile. Often, he played bankers (he was one of Potter’s minions in It’s a Wonderful Life, and is best known as Homer Bedlowe in Petticoat Junction, who kept trying to put the railroad out of business.
Since everyone seems to be naming guys, let me also add that my favorite female character actor is the late (very late, died in 1942) Edna May Oliver. She was the sniffy dowager in dozens of '30s movies, including my favorite of her roles, that of Lady Catherine DeBurgh in the Greer Garson-Lawrence Olivier version of Pride and Prejudice, one of my favorite movies. The snapshot picture in the link doesn’t do her justice, but I bet if most of you saw her on screen, you’d know her immediately, if you like old movies.
. . . too many to choose from to name one “favorite.” In addition to the already-mentioned Edna May Oliver, Edward Everett Horton and Thelma Ritter, I’ll add Margaret Dumont, Jimmy Durante, Alice Brady, Skeets Gallagher, Billie Burke, Una Merkel, Vince Barnett, and the great “floozies,” Iris Adrian, Jean Carson, Barbara Nichols, Joan Shawlee . . .
I got timed out before I posted my own picks: J. K. Simmons, the real star of the Spider-Man movies, and - if he can qualify as a character actor - Joseph Cotten.
I agree with many of the people mentioned - Steve Buscemi, John Hurt, William H Macy, Pete Postlethwaite and others - but I haven’t seen John Turturro mentioned. He immerses himself in his roles like very few actors can. I can’t believe that Barton Fink is played by the same guy as Joey Knish from Rounders and John Shooter from Secret Window.
John is indeed a marvelous actor. He might be to much of a star to really qualify as a character actor
I was gonna say “naaah”, as when I think of him I think of secondary characters that steal the scenes they’re in and that I couldn’t think of a starring role for him, but then I realized I had myself mentioned a movie in which he plays the leading role - Barton Fink.
John Doucette (short, burly)
Neville Brand (stocky, greasy)
Olan Soule (skinny, nerdy)
Steve Ihnat (serious, died-way-too-young)
and dammit, what’s his name, was on “Daniel Boone,” killed his wife and himself in the late 80s in a murder/suicide pact, dammit… somebody help… great whiny voice…
Sir Rhosis
^^^Albert Salmi… of course I remember his name as soon as I reply.
SR
Lot’s of great nominations here. It’s amazing how often we see some of these folks in films and on TV but never really know who they are.
I’m glad to see someone mentioned Ben Johnson He was always a favorite of mine.
How about Harry Morgan
Even though he had some costarring roles the bulk of his work would be as a charecter actor.
Ray Walston is another name that comes to mind.
Also LQ Jones
Jeanette Nolan who played Lady McBeth in the Orson Wells version but I remember a remarkable charecter named Dirty Sally on some Gunsmoke episodes
How about some ethnic charecter actors and actresses? People who stay busy playing ethnic roles.
Neville Brand …great call. I loved him in a lot of roles but most of all as Lightfoot, the half breed tracker with John Wayne in Cahill US Marshall. He also starred as Reese Bennett Texas ranger in the short lived TV show Laredo, with another favorite tuff guy charecter actor of mine
William Smith It looks like he’s doing a bunch of crappy horror movies now {in his seventies} You might remember him slugging it out with Clint Eastwood in 1980 in Any which way you can.
As the first offering for ethnic charecter actors I give you Mako
Clint Howard rules the character actor world.
That would be Harriet Sansom-Harris, and I agree. Jayjay and I were lucky enough to see her as “Vera Charles” in a production of “Mame”, last year. The star of the show was another one of my favorite character actor/actresses, and another “Frasier” alum Christine Baranski.
Needless to say the show was terrific. 
As a fan of the old Universal movies, there will always be room in my heart for George Zucco, Lionel Atwill, Dwight Fry and Evelyn Ankers.
J.T. Walsh and a vote too for Chris Cooper.
I found a site about character actors and actresses here
MAn oh MAn. A lot of familiar faces there.
I was specifically looking for Allan Melvin a long time staple of TV in the sixties.
Can’t believe no one has mentioned Keith David. He’s been in some great movies and is almost always perfect.
Dunno if he’s “ethnic” per se, but Bernie Koppell worked for a long time playing mexican characters, of all things.