This was a business establishment that allowed the gambling on stocks or commodities. The bets often included a high degree of leverage.
What separated a bucket shop from traditional brokerage firms is the lack of intention to deliver the stock or commodity. The practice was to get people speculating on price, hoping for appreciation.
These appeared in the United States in the mid 1800s but most disappeared by the 1920s. The practice was outlawed by many states, spreading over time.
The problem was that the firms had a conflict of interest. Their goal was to lure in more people. So when an individual made a bit of money, that was promoted and advertised. Losses, however, were also to the profit of the bucket shop.
Since there is no intention to buy the securities, the brokerage activity is actually non-existent. Even boiler room operations do engage in actual trading of securities, albeit with deception and often using fraud.
Legality
Bucket shops claimed to use derivatives as a basis for the trades. There was no trading going through exchanges, meaning that customers were simply betting on the price going up or down without having anything in return. With a stock, even if the price does drop, the shareholder has certain rights. Also, since it is traded on an exchange, the asset can be sold for whatever market value is.
In this instance the bucket shop is playing the part of the bank, taking the other side of the bet against its customers.
This practice is not only illegal for bucket shops but also legitimate brokerage firms.
In 1922, the state of New York passed the Martin Act which effectively made bucket shops illegal.