The phrase top banana denotes the most important person in an organisation or activity.

Its original meaning was, in American theatre slang, the leading comic in a burlesque entertainment.

More broadly, banana, preceded by a word indicating the ranking, denoted a stage comedian having a specified position within a team or act. This is clear from the text in which is the earliest instance of top banana that I have found: an article by Charles D. Rice about the American comedian Phil Silvers (Philip Silversmith – 1911-85), published in The Cincinnati Enquirer (Cincinnati, Ohio) of Sunday 28^th December 1947—top banana is used as the title, but in the article itself, the word that is used is first banana, in contrast to second banana and third banana:

Top Banana . . .

Three days after “High Button Shoes,” Broadway’s new musical hit, opened, Phil Silvers was still swamped with telegrams of congratulations. He read and re-read them with hoarse murmurs of gratitude (he had laryngitis). One in particular brought forth croaks of delight every time he unfolded it: “LOVE AND KISSES, RITA HAYWORTH.” “Not bad,” he said, “for a guy who started out as Third Banana at Minsky’s.”
Third Banana, for the uninitiated, is a very humble station in show business. During the depression Phil played burlesque; the average burlesque company carries three comedians who are always known as First, Second and Third Bananas. (Non-comedians are Straight Men, chorus girls are Slaves, and any female performer who is articulate enough to speak lines has the dignified title of Talking Woman […])
[…]
Actually, Phil didn’t start in show business as a Third Banana. He really began at the age of eight in the tough Brownsville section of Brooklyn, doing a singing act with playmates on street corners and in bars.
A few years later Gus Edwards heard him on the boardwalk at Coney Island, recruited him for his famous kiddie review. Phil worked with Edwards for three years. Then his voice changed and he got fired. He tried another vaudeville act as a juvenile comedian. After a couple of years, vaudeville died (he wasn’t specifically responsible). Finally he went into burlesque. Success! Third Banana . . . Second Banana . . . First Banana!

The contrast had appeared earlier, in Burlesque Gets More Elegant, Even in Boston, published in the Fort Myers News-Press (Fort Myers, Florida) of Friday 31^st January 1947:

New York, Jan. 30—(AP)—Burleque [sic], for years the blousy, brash, black sheep of show business, is going elegant. At least, Eddie Lynch says it is. Eddie, an old timer who is a producer for the eastern burlesque wheel, says succinctly: “The days of bums bringing in their lunch and taking their shoes off is [sic] over.”
[…]
There’s been a delicate shift of emphasis from flesh to comedy. In the salt slang of the trade Lynch explains: “We’re featuring comics. We book a first and second banana [sic], you know, comics; a ‘yes’ man (straight man); two strippers and a featured strip, the dame who rides in the pullman.”

The second-earliest instance of top banana that I have found also appears in contrast to second banana and third banana; it is from the column Up and Down Broadway, by Jack Gaver, published in The Delta Democrat-Times (Greenville, Mississippi) of Wednesday 11^th August 1948:

One of the unsung heroes of the summer season on Broadway was Joey Faye of “High Button Shoes.”
[…]
Joey, who is no ancient, was fortunate enough to get into burlesque a little before the final phase that brought its demise in New York and changed it for the worse in those places where it survives. Some of his contemporaries in burlesque were the late Rags Ragland and Abbott and Costello of the movies. Joey was a “top banana” (burlesque for comic) at a time when A. & C. were second and third bananas. When burlesque became the realm of stripteasers and comics were simply tolerated, Joey went “legit.”

According to the following from About: Bananas, by John Wilcock, published in The New York Times (New York, N.Y.) of Sunday 30^th March 1958, top banana originated in a skit involving the sharing of a banana:

[Comedian Phil] Silvers, incidentally, credits another burlesque comedian, Harry Steppe*, with introducing the phrase “top banana” into show business jargon in 1927 as a synonym for the top comic on the bill. It rose out of a routine, full of doubletalk, in which three comedians tried to share two bananas.

(* Harry Steppe (Abraham Stepner – 1888-1934))

But, even granting that the phrase top banana (like old chestnut for example) was confined to the theatrical profession for several years, it is surprising that it did not appear in print until 1947, twenty years after it was supposedly coined—and that this explanation appeared only in 1958.

Other words or phrases of theatrical origin:
to be decent (sufficiently clothed to see visitors)
malapropism
claptrap
cloak-and-dagger
Mrs Grundy
Paul Pry
Mummerset
simon-pure
fedora
Box and Cox
Hamlet without the Prince
slapstick
to play to the gallery
to steal someone’s thunder
bums on seats
P’s and Q’s
something is rotten in the state of Denmark
bob’s your uncle