The American-English phrase behind the eight ball means in trouble or at a disadvantage.

The terms eight ball and eight rock denote the black ball, numbered eight, in the North-American variety of pool, and, by extension, this variety of pool.

(This variety of pool uses two sets of seven coloured or patterned balls, together with one black ball and one white cue ball, the aim being to pocket all of one’s own balls followed by the black.)

Both eight ball and eight rock have come to be also used derogatorily and offensively to denote a black person—as illustrated by the following from the Jackson Daily News (Jackson, Mississippi) of Monday 27^th November 1911:

It is a well known fact that the colored population resents, as a rule, the word “nigger” being applied to them, but the height of their resentment is reached when the term “black nigger” is applied, especially if it comes from one of their own race.
At a recent trial before Judge Manship a large black African named John Adams was arraigned before the court on a charge of assaulting Will Harris, a gingercake “complected pusson,” by beating him over the head with a billiard cue, much to its damage (the cue, of course).
[…]
“What language did he use that so aroused your ire?” queried the judge.
[…]
“Jedge, ah jest natur’ly hates ter use sich lan’widge in dis yere cote, but he—he—called me a eight rock.”
“An eight rock! what in the name of the Seven Sunderland Sisters do you mean?”
“Why, jedge, ain’t you neveh played pool wed striped an’ colored balls? De eight rock, jedge, is de black ball.”
“Well of all the—how black is the eight ball, anyhow?”
“Hits blacker dan a black cat in a dark cellar at midnight.”

The earliest occurrence of the phrase behind the eight ball that I have found is from one of the unconnected paragraphs making up the column On the Fence, published in The Daily Hammer—itself published in the Buffalo Evening Times (Buffalo, New York) of Tuesday 25^th September 1923:

If Charley Heinz doesn’t stop buying stocks he may wind up behind the eight ball.

The second-earliest occurrence that I have found is from the column On “the Trail” of Sport, published in the Hamilton Evening Journal (Hamilton, Ohio) of Thursday 19^th August 1926:

The Phillies ended their season up “behind the eight ball,” as the vulgar saying goes. The Quakers won 2 of 11 games played. The series total now stands 15 to 4 in favor of Cincinnati.

The phrase then occurs in the column Runyon Says, by Damon Runyon, published in The Evening News (Harrisburg, Pennsylvania) of Monday 18^th October 1926:

Herrera was a tremendous hitter for a little man. He was a featherweight, and he gave Terrible Terry McGovern one of the hardest battles of his career. But Herrera was no hand for training, and he wound up behind the eight ball, as the boys say.

The phrase behind the eight ball refers to a position in a game of pool in which a player cannot make a direct shot at the target ball because the black eight ball obstructs the cue ball—as explained by Harold Hadley in Behind The Eight Ball, a short story published in the Daily News (New York City, N.Y.) of Wednesday 4^th December 1929:

“Time for a game of pay ball,” said Tommy. “Fifty cents a ball. I’ll take the six and ten, you take the five and nine. All right?”
“Sure,” said James. “It’s a bet.”
They played. Quickly the balls were run down to the five, and Tommy’s last shot missed James’ five ball. If he sank this one, which stopped near a pocket, James was to win 50 cents. But when he set his cue between his thumb and fingers and took aim he saw a black ball between the cue ball and the five. He was behind the eight ball, and on the bank shot, he missed, Tommy sinking the five.
Ironically enough, when he was shooting at Tommy’s ball, again the eight ball was in the way, a sort of pool stymie.
James cursed and missed a bank shot. So the game went; when he was shooting for money, he was behind the eight ball.

Illustration for Behind The Eight Ball—Daily News (New York City, N.Y.)—4^th December 1929:

When he was shooting for money, he was behind the eight ball.