Overview of Batman & The Joker's Origins
Sources:
Wikipedia
Entertainment Weekly Bob Kane Interview, 22 July 1994
History of Comic Books In America, by Les Daniels

In early 1939, DC's success with the seminal superhero Superman in Action Comics prompted editors to scramble for more such heroes. In response, Bob Kane conceived "the Bat-Man". Kane said his influences for the character included actor Douglas Fairbanks' movie portrayal of the swashbuckler Zorro, Leonardo Da Vinci's diagram of the ornithopter, a flying machine with huge bat-like wings; and the silent film The Bat Whispers (1930), based on Mary Rinehart's mystery novel The Circular Staircase.

Bill Finger joined Bob Kane's nascent studio in 1938. He recalled that Kane

“ had an idea for a character called 'Batman', and he'd like me to see the drawings.

Finger offered such suggestions as giving the character a cowl and scalloped cape instead of wings; adding gloves; leaving the mask's eyeholes blank to connote mystery; and removing the bright red sections of the original costume, suggesting instead a gray-and-black color scheme. His suggestions were influenced by Lee Falk's popular The Phantom, a syndicated newspaper comic strip character with which Kane was familiar as well. Finger, who said he also devised the character's civilian name, Bruce Wayne, wrote the first Batman story, while Kane provided art.

Kane, who had already submitted the proposal for Batman at DC, and held a contract, is the only person given official company credit for Batman's creation.

According to Kane:

“ Bill Finger was a contributing force on Batman right from the beginning. He wrote most of the great stories and was influential in setting the style and genre other writers would emulate ...I made Batman a superhero-vigilante when I first created him. Bill turned him into a scientific detective.”

The character debuted in Detective Comics #27 (May 1939) and proved a breakout hit.

Batman's archnemesis the Joker was introduced shortly afterward, in Batman #1 (Spring 1940).

Of the Joker, Kane says:

“ Bill Finger and I created the Joker. Bill was the writer. [The Joker] looks like Conrad Veidt — you know, the actor in The Man Who Laughs, the 1928 movie based on the novel by Victor Hugo. ... Bill Finger had a book with a photograph of Conrad Veidt and showed it to me and said, 'Here's the Joker'."