Andrews was born in Beloit, Wisconsin, and was educated there at Beloit College. After graduating, he took up a post at the American Museum of Natural History, New York. His early interest lay in whales and other aquatic mammals, and these he collected assiduously on a number of museum-sponsored expeditions to Alaska, North Korea, and the Dutch East Indies (Indonesia) between 1908 and 1913 It was largely through Andrews's efforts that the collection of cetaceans at the American Museum of Natural History became one of the most complete in the world.

Andrews is best known for his discovery of previously unknown Asiatic fossils. Most of his findings were made on three expeditions to Asia, which he led as chief of the Asiatic Exploration Division of the American Museum of Natural History. The first of these was to Tibet, southwestern China, and Burma (1916–17); he then visited northern China and Outer Mongolia (1919), and central Asia (1921–22 and 1925). The third Asian expedition produced major finds of fossil reptiles and mammals, including remains of the largest known land mammal, the Paraceratherium (formerly called Baluchitherium), an Oligocene relative of the modern rhinoceros, which stood some 17–18 feet (5.5 m) at the shoulder. In Mongolia, Andrews discovered the first known fossil dinosaur eggs. He was also able to trace previously unknown geological strata, and unearthed evidence of primitive human life on the central Asian plateau.

Andrews was appointed director of the American Museum of Natural History in 1935, but resigned in 1942 in order to devote himself entirely to writing about his travels and discoveries.