Boscov was born in Reading, Pennsylvania in 1929, the son of Russian Jewish immigrants Solomon and Ethel Boscov, His father had arrived in Reading in 1911 where he founded a dry goods store in a row house in 1918.
Albert graduated from Reading High School and Drexel University. It was at Drexel where Boscov started his first business, a sandwich delivery service called U-Eat-Em. Boscov went on to serve in the Navy during the Korean War. After returning from military service, he returned to Reading and joined his father's business.
In 1959, he married his wife Eunice, with whom he had three children.
After taking over the family business, Boscov expanded the original store at Ninth and Pike Streets in Reading. In 1962, he opened Boscov's West in Sinking Spring, Pennsylvania, the company's first full-line store. Boscov's East opened in Exeter, Pennsylvania in 1967. Throughout the 1970s and 1980s, Boscov and his brother-in-law, Edwin Lakin, expanded the chain throughout Pennsylvania, Maryland, Delaware, New York and New Jersey. By 2005, Boscov's had 39 stores with $1 billion in sales.
In early 2006, Boscov retired as chairman transferring control to his nephew, Ken Lakin, who soon after purchased a 10 department store chain by borrowing heavily.
In August 2008, Boscov's filed for Chapter 11 Bankruptcy, citing deteriorating consumer and credit conditions due to the economic crisis of 2008. In October 2008, Albert Boscov and his brother-in-law Edwin Lakin both confirmed that they would be making a bid to purchase back the chain, competing against Versa Capital Management, Inc.. On November 4, 2008, it was confirmed that Boscov's would be selling its assets back to Albert Boscov and Edwin Lakin, canceling the earlier agreement with Versa, and keeping the 98-year-old chain under family control. In March 2015, he retired for the second time and handed control of the company to his nephew, Jim Boscov.
In 2010, Politics Magazine named him one of the most influential Democrats in Pennsylvania.
Boscov was a philanthropist who donated money to Penn State Berks campus for its library, and to Genesius Theatre for restoration and further improvement of its main-stage building in Reading, Pennsylvania. He also performed at Genesius Theatre as Tevye from Fiddler on the Roof at Genesius's "Broadway Musical Revue". Boscov was an Ashkenazi member of the Jewish community of Reading, Pennsylvania.
He died on February 10, 2017 from pancreatic cancer in Reading, Pennsylvania.