Wassell was the son of John Wassell, and English-born lawyer who was elected mayor of Little Rock in 1868. At about the same time, Albert Wassell began playing baseball, as did his brother Samuel. Albert played with the Dixie Base Ball Club from 1868-1969, as well as for St. John's College, presumably as a student. He later attended the University of Michigan.
In 1875, Wassell and his brother joined the Accidentals Base Ball Club. The team participated in the ((Foul Flag Association (1875)|Foul Flag Association), winning the championship. In November, Wassell and the Accidentals played a series of two exhibition games with the St. Louis Red Stockings, a team which had been a part of the professional National Association early that year. The Red Stockings won both games.
In 1877, Wassell married Leona McAlmont and together they had three children. Their second was Dr. Corydon Wassell, who became famous for his work as a medical missionary and his heroics during World War II. In 1944, the film The Story of Dr. Wassell was released and starred Gary Cooper, who is best known in the baseball world for potraying Lou Gehrig in the film The Pride of the Yankees.
Albert Wassell lived in Little Rock almost his entire life, working as a lawyer. He died there December 18th, 1938 and was buried at Mt. Holly Cemetery.