Virtually any of Donald E. Westlake’s crime novels, especially the Dortmunder series. Wonderful capers, excellent plotting, great characters, and also vastly amusing. There’s no need to start with any particular one, but my favorites are What’s the Worst the can Happen, Good Behavior, Drowned Hopes, and Don’t Ask. All of them are crime novels, not mystery novel.
I ADORE Robert Crais. And Jonathan Kellerman. Reginald Hill (Yorkshire England police).
How about Ed McBain? He wrote a zillion books about a (made-up) NYC police station and the people manning it. If you liked NYPD Blue, Ed McBain’s books will be just your speed.
No love for Dennis Lehane? He seems to be one of Hollywood’s favorites - Mystic River and Gone, Baby, Gone were both adapted into very good movies, and another one, Shutter Island is forthcoming. I haven’t read all that much of his stuff, but I’ve greatly enjoyed what I have - Mystic River and Shutter Island among them. Those two are stand-alone works; Gone, Baby, Gone takes place in the series involving private investigator partners Patrick Kenzie and Angela Gennaro, of which I’ve only read the first, A Drink Before the War, which I thought was very good.
This one works only if you have not seen the movie (which doesn’t work at all): A Kiss Before Dying. Very simple, but an oldie.
I’ve greatly enjoyed The Man Who… books by Stephen R. Donaldson.
The Man Who Killed His Brother (1980)
The Man Who Risked His Partner (1984)
The Man Who Tried to Get Away (1990)
The Man Who Fought Alone (2001)
Yes, this is the same SRD who wrote the Thomas Covenant books, but there aren’t any similarities at all. His writing style is different, the outlook of his characters is different, etc.
These are hard-boiled, talking-with-fists types of stories. Bad things happen with alarming frequency to the good guys, and solving a case doesn’t always bring any kind of closure to the events in the book.
Plus, for most of them, it’s very difficult to see the end before you get there, if you know what I mean.
Another vote for Conan Doyle’s Sherlock Holmes stories. “The Red-Headed League,” “The Speckled Band,” “Silver Blaze,” “The Greek Interpreter,” “The Norwood Builder,” “The Musgrave Ritual,” “The Blue Carbuncle” and “The Second Stain” are among my faves. Also, of course, you must read the Holmes novel The Hound of the Baskervilles. Clever mysteries, a compelling lead character, and Victorian atmosphere out the whazoo.
Another vote also for the Spenser books (although I’ve missed the last few). Fun, more-or-less contemporary American detective stories.
I really enjoyed Gun Monkeys by Victor Gischler, about a crew of assassins in the midst of a gang war after their boss gets whacked. Funny, bloody and a real page-turner.
**Micheal Connelly’s **books, especially “The Brass Verdict” with his protagonist, Det. Hieronymous Bosch, are fun reads.
**Henning Mankell **from Sweden writes as good a crime novel as anyone. The Kurt Wallender series are great.
Second James Lee Burke.
**Elmore Leonard’s **writing is tight and real.
Lee Child’s Reacher series is fun and fast
Steig Larsson’s The Girl with the Dragon Tattoo is (in translation, at least) an easy read, and a visceral and intriguing mystery novel. It’s the first in a trilogy.
Well, I’m afraid I’m going to be only half a help. I remember reading an extraordinary mystery story, but I remember neither the title, nor the author.
Here’s how it went : the scene was set during the early Renaissance, inside of a British college if I’m not mistaken. The mystery was told through four sets of eyes. The first narrator tells the bare bones of the story - sets the scenery, the principal characters, etc… At the end of that story, a second narrator steps in, says “Well, this first narrator got it mostly right, but he lied a lot too, cause this is how it happened…”, and so on, for a third, then a fourth narrator. Even at then end of the fourth narration, you’re not exactly sure what really happened, on account of the multiple plots that happened all at once. Kind of a British Roshambo. Any Doper feel like giving away the title and author, feel free. As I recall, it was “something… of the Cross”. Then again, I read the French translation so the title"s translation might well be a full thesaurus away from the actual title 
Sorry to be of so little help. Once I’m sober, I’ll find the book I promise.
Should also have mentioned Robert Harris’s Fatherland, an alternative-history detective story set in the Third Reich in 1964. An SS criminal investigator looks into the mysterious deaths of several senior Nazi Party officials just before Hitler holds a summit meeting with President Joseph Kennedy. An intriguing and smart novel.
For fast enjoyable “detective” novels, you can’t go wrong with the late Gregory MacDonald’s Fletch series (and the three Flynn books, kind of a spin-off from the former.)
Try to read them in order (although I think they’re all out of print. Maybe you can find them in a library or on PaperbackSwap.com).
“An Instance of the Fingerpost” – Iain Pears?
Carl Haaisen is colorful, fastpaced and pretty funny. If you hear Florida jokes over there in England and want to know what’s behind it, look no further.
Avoid, at all cost, Janet Evanovich’s Stephanie Plum series. Unless you like reading books in which you’d like to choke the crap out of the main character. Plum is the epitome of stupid git.
Fletch is outstanding; the best ending of anything I’ve read. The others are good, but not nearly as good as the first. Perfection cannot be equaled, nor even approached. 
Too late for edit. I also enjoyed 2 of Jonathan Kellerman’s Alex Delaware novels: When The Bough Breaks and Private Eyes.
I loved the film of Gone Baby Gone - Amy Ryan was fantastic. I might check the series out.
That’s a little unfair. In the book, it was very clear who did it. However, the movie changed things around because the chauffeur, in the book, was murdered by the Lauren Bacall character. The movie, of course, couldn’t have that, so switched around enough to make it thoroughly confusing. Howard Hawks, the director, said he wasn’t sure who killed whom either, but it’s a great movie anyhow. It’s not a whodunit, it’s a hard-boiled film noir with Bogart and Bacall, what more could one ask for?
I agree, Raymond Chandler is the place to start. He invented the genre (although Dashiel Hammet helped, but I think Hammett is inferior in almost every way.)
It was?
It is.
The Library of America produced two great collections, American Noir of the 1930s and 40s and American Noir of the 1950s. The novels included are:
The Postman Always Rings Twice - James M. Cain
They Shoot Horses, Don’t They? - Horace McCoy
Thieves Like Us - Edward Anderson
The Big Clock - Kenneth Fearing
Nightmare Alley - William Lindsay Gresham
I Married a Dead Man - Cornell Woolrich
The Killer Inside Me - Jim Thompson
The Talented Mr. Ripley - Patricia Highsmith
Pick-Up - Charles Willeford
Down There - David Goodis
The Real Cool Killers - Chester Himes