He was born in Philadelphia, Pennsylvania, the son of Ezekiel and Sarah (Howell) Abbett. He attended Central High School and graduated in 1853. He was admitted to the Philadelphia bar in 1858. In April 1861, he moved to New York to open a law firm and formed a partnership with William Fuller. Abbett married Mary Briggs of Philadelphia in 1862 and the couple moved to Hoboken, New Jersey.
In 1864, Abbett entered politics and was elected as a Democrat to the New Jersey General Assembly, where he served from 1864-1866; and 1869-1870 when he was also the Speaker. He also served one term in the New Jersey Senate from 1875-1877 and was Senate President in the '77 session. Abbett was elected governor of New Jersey in 1883 (defeating Jonathan Dixon in '83 and Civil War hero E. Burd Grubb in '89) and served two non-consecutive terms from 1884-1887 and again from 1890-1893.