Burgh was born into a rich brewer's family. He studied medicine in Leiden in 1614 and became a doctor in 1618 in Amsterdam. In the same year he entered the city council as a Calvinist. He changed his view within a couple of years, paying a fine for the famous Dutch poet Vondel. Vondel had gotten into trouble because of his play Palamedes, in which he was recalling the beheading of Johan van Oldenbarneveldt. Oldenbarneveldt had pleaded for peace with Spain and for shrinking the state army.
In 1629 Albert Burgh owned land in Rensselaerswyck, Albany, which he sold to the main investor Kiliaen van Rensselaer. He helped famous Dutch scholar Caspar Barlaeus with a teaching job at the Athenaeum Illustre after he had been fired in Leiden in 1619. In 1638, as one of Amsterdam's four Burgemeesters, Albert Burgh offered Marie de' Medici a meal with rice, in those days very exotic and hardly known to Europeans. He sold her a famous silver rosary, captured in 1629 by Piet Hein in Brazil.
During his lifetime he visited Moscovia twice, in order to improve trade relations. Burgh died on Christmas Eve in Novgorod. The corpse was returned to Amsterdam. Dirck Tulp, the son of the famous surgeon Nicolaes Tulp who had accompanied him on his trip to Moscovia, married his daughter Anna.
One of Albert Burgh's grandsons, also named Albert Burgh, was a Franciscan in Rome and argued with his former teacher Baruch Spinoza in a couple of curious and famous letters; another grandson of Albert Burgh was the mayor of Amsterdam Coenraad van Beuningen.
- Elias, J.E. (1903–1905, reprint 1963) De vroedschap van Amsterdam 1578-1795, two volumes.