Renee Montoya is an interesting case. She was created by the writing staff of Batman: The Animated Series, and first appeared in the episode "Pretty Poison" (September 1992). However, she first appeared in comics in March 1992 in Batman #475. This is due to production delays on the animated series.
Since her introduction, Montoya has become a very popular character. Since her debut, she has been revealed to be a lesbian, been one of the lead characters in a comic (Gotham Central), and taken over the role of The Question, after which time she began dating Batwoman and got a regular back-up feature in Detective Comics.
Nora Fries first appeared in the Batman: The Animated Series episode "Heart of Ice" (1992) as a part of Mr. Freeze's new origin. Her first comics appearance was Batman: Mr. Freeze (1997).
As part of an agreement, Nyssa al-Ghul allowed Mr. Freeze to put Nora Fries in a Lazarus Pit; however, her body was do damaged from cryogenesis that it turned her into Lazara (first appearance: Batgirl (vol. 2) #70 (January 2006)), a villain that can generate flame and revive the dead.
Ferris Boyle first appeared in the Batman: The Animated Series episode "Heart of Ice" (1992) as a part of Mr. Freeze's new origin (he was Victor Fries's boss). His first comics appearance was Batman: Mr. Freeze (1997).
Summer Gleeson first appeared in the Batman: The Animated Series episode "Heart of Ice" (1992). Although she hasn't had a proper first appearance, she received a Gotham Gazette byline in Batman: No Man's Land - Secret Files and Origins #1 (December 1999) and appeared on tv in Batman: Gotham Knights #33 (November 2002), although this was a non-canon back-up story.
The Gray Ghost first appeared in the Batman: The Animated Series episode, "Beware the Gray Ghost" (1992), as a noir serial character that Bruce Wayne loved to watch as a kid. Despite being a fan favorite character, it took him almost 20 years to make the jump to comics, and even then, it was in a highly changed form. In Batgirl #15 (January 2011), the Grey Ghost appears as Clancy Johnson, Batgirl's self-proclaimed guardian angel...who has a secret. (Clancy based his secret identity on the Gray Ghost tv show, which now exists in DC Comics).
The Mad Bomber first appeared in the Batman: The Animated Series episode, "Beware the Gray Ghost" (1992), as a prop-making terrorist villain of the Gray Ghost. His first comics appearance was in Batgirl #9 (June 2010), and later it was revealed that he was actually Clancy Johnson, who based both of his alternate personas on characters from the Gray Ghost tv show.
Harley Quinn first appeared in the Batman: The Animated Series episode "Joker's Favor" (1992). Her first canon appearance in comics was in Batman: Harley Quinn #1 (October 1999), though her actual first appearance - not counting Batman Adventures - was in Batgirl + Batman: Thrillkiller '62 (October 1998) as Hayley Fitzpatrick, the girlfriend of Joker analogue Bianca Steeplechase.
She is considered the poster child for canon immigrants and is one of the most well known and successful, making appearances in Birds of Prey, The Batman, Lego Batman, Batman: Arkham Asylum, and others. (The only canon immigrants considered more successful are the ones most people don't even know are canon immigrants, such as Perry White and Jimmy Olsen.)
The Clock King
The Clock King is another of those interesting cases, because he actually started out as a comics characters; or, at least, a version of him did. The Clock King first appeared in World's Finest Comics #111 (August 1960) as William Tockman, a villain who used clock-related gadgets in his crimes.
The Clock King that first appeared in Batman: The Animated Series, though (the episode "The Clock King" (1992), to be precise), is Temple Fugate, a guy who is obsessed with time and punctuality. He first appeared in comics in Teen Titans #57 (May 2009) as a member of the Terror Titans. This Temple Fugate has "absolute time sense".
Sewer King first appeared in the Batman: The Animated Series episode "The Underdwellers" (1992). His only appearance in comics to date was in 52 #25 (October 2006).
Condiment King has moved to his own page.
Salvatore Valestra (aka Chuckie Sol)
Salvatore Valestra and Chuckie Sol appeared as two mob bosses in Batman: Mask of the Phantasm. Their first comics appearance was in Justice League of America 80-page Giant #1 (November 2009) as a WWII-era mob boss named Valestra (Chuckie Sol was an alias).
Roxy Rocket actually first appeared in Batman Adventures Annual #1 (January 1994), a comic series based on Batman: The Animated Series and set in its continuity. She later appeared in the series proper in the episode "The Ultimate Thrill" (1998). Her first comics appearance was a cameo in Detective Comics #822 (October 2006), though her first full appearance was in Batgirl (vol. 3) #6 (March 2010).
Farmer Brown has moved to his own page.
It wasn't just characters, though. Several characters' designs made the jump as well:
Robin (Dick Grayson)
Left: Dick Grayson as Robin
Middle: Dick Grayson as Robin on Batman: The Animated Series (1992)
Right: Dick Grayson as Robin in "Nightwing: Year One" (Nightwing #101, March 2005)
Clayface (Basil Karlo) was originally just a guy that could change the shape of his face (Detective Comics #40, June 1940). Later, there was Matt Hagan, who was a man made out of clay. In Batman: The Animated Series, though, he became a monster made out of clay, which inspired the appearance of a new Clayface, Cassius Payne (the son of Basil Karlo and Sondra Fuller, another Clayface), who first appeared in Batman #550 (January 1998).
Robin (Tim Drake)
Left: Tim Drake's first comics costume
Middle: Tim Drake on The New Batman Adventures
Right: Tim Drake's One Year Later costume (the colors are in memory of Superboy), starting with Teen Titans #34 (May 2006)
Mr. Freeze (alternate reality)
Left: How Mr. Freeze looked in the New Batman Adventures episode, "Cold Comfort" (1997)
Right: How Mr. Freeze looked during Flashpoint: Citizen Cold #1 (August 2011)
As well as certain storytelling elements:
Killer Croc's condition
When Killer Croc first appeared, he was noted as having a rare skin condition. In Batman: The Animated Series (1992), however, his appearance was caused by a mixture of human and reptile DNA. I don't know yet the first time it changed, but in Batman #608 (December 2002), his condition was said to be regressing, causing him to gain a pronounced jaw and a tail.
Batman and Zatanna's former relationship/Batman as a former trainee of Zatara
These two plot points first appeared in the Batman: The Animated Series episode "Zatanna" (1993) and later came to light in Detective Comics #824 (December 2006) and #833 (August 2007), both of which were written by Paul Dini.