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. Joost van den Vondel got into trouble because of a play Palamedes, in which he was reminding to the beheading of Johan van Oldenbarneveldt. Oldenbarneveldt pleaded for peace with Spain and 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 Caspar Barlaeus, a famous Dutch scholar with a teaching job at the Athenaeum Illustre, after he was fired in Leiden in 1619. In 1638 he and Andries Bicker 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.
Grain was extremely important for the city's wealth and influence. 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 Moscow, married his daughter Anna.
Albert Burgh, a Franciscan in Rome, and one of his grandsons, argued with his former teacher Baruch Spinoza in a couple of curious and famous letters, and another was the mayor Coenraad van Beuningen.
- Elias, J.E. (1903–1905, reprint 1963) De vroedschap van Amsterdam 1578-1795, two volumes.