Argelander was born in Memel in the Kingdom of Prussia (now Klaipėda in Lithuania), the son of a father of Finnish descent, Johann Gottlieb Argelander, and German (Prussian) mother, Dorothea Wilhelmina Grünhagen.

There he designed and built a new observatory at the University of Bonn with funding approved directly by King Frederick William IV whom Argelander had become friends with in his childhood.

This lifelong friendship had started when the then crown prince temporarily lived in Argelander's parents house after the Prussian royal family fled to Memel after the Battle of Jena–Auerstedt during the Napoleonic Wars.

Argelander excelled in developing effective, simple and fast methods for measuring star positions and magnitudes, thereby making a pioneering work for modern astronomy.

In 1863, Argelander helped lead in the founding of an international organization of astronomers named the Astronomische Gesellschaft.