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.
Around 1624 Burgh became one of the managers of the Dutch West India Company and owned land on the New Jersey side opposite the river Delaware. In 1632 Albert Burgh sold his land in Rensselaerswyck, Albany, to the main investor Kiliaen van Rensselaer.
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. In 1644 he became a manager of the Admirality of Amsterdam.
During his lifetime he visited Moscovia twice (1629 and 1647), in order to improve trade relations. Both times he enterened the country in Archangelsk. 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. In 1652 Fort Coenraadsburg on the Gold Coast was named after him.
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.