Henry Carr Pritchett was born on August 12, 1852, in Warren County, Missouri, to William Ira and Martha Ann Hubbard. His younger brother, Joseph (Joe) Lucien Pritchett, was a mathematics professor and Dean at Sam Houston Normal Institute.

In 1867, his family moved to Glasgow, Missouri, where Henry attended Prichett School Institute, a private school founded in 1866, named after his uncle, Carr Waller Prichett, the Institute's first principal. Henry graduated from the Institute in 1873 and received his master's in arts at the Institute two years later. Afterward, he continued his education at the University of Michigan in graduate studies.

On December 28, 1876, Henry married Kate Smith in Missouri. They moved to San Marcos, Texas, where he was elected as a teacher and co-principal of the Coronal Institute. Three years (1881) later, they moved to Huntsville, Texas, for his new role as a mathematics and astronomy professor at Sam Houston Normal Institute. On June 12, 1884, they had a son named Henry Lucien Pritchett. In 1890, Henry became the state superintendent of instruction until his new position as president in 1891.

The Texas Governor, Sul Ross, appointed Henry Pritchett as the Institute's president (the first time the term "president" was used instead of principal) and was approved by the current president, Joseph A. Baldwin, upon his resignation to become the new chair pedagogy at the University of Texas. Pritchett's presidency at SHNI lasted for seventeen years (1891-1908), where he saw the construction of the Peabody Memorial Library, the first separate library at a Texan college.

Henry Carr Pritchett died on March 9, 1908, in Galveston, Texas. He is buried in San Marcos, TX.