Buel was born in Castleton, Vermont and exhibited precocious intellectual abilities. He attended the public schools in Poultney, Vermont until the age of eight, at which time he began an accelerated course of studies preparing for college under the private tutelage of Henry Howe, the former principal of the Canandaiga Academy in Canandaigua, New York. In 1824, at the age of 10, he passed the admission exams for Middlebury College in Middlebury, Vermont and was admitted to the college. However, concerned about the difference in age between Buel and the other college students, his teacher and family persuaded him to continue his independent studies for three years. In 1827, at the age of 13, Buel accompanied one of his teachers and was enrolled as a sophomore at the University of Vermont in Burlington, where he stayed for one year. In 1828, he decided to attend Middlebury College, which was his original intent. He was enrolled there as a junior and also taught at the common schools in Clarendon.

In August 1830, at the age of 16, Buel graduated from Middlebury College ranked first in his class. In September 1830, he became superintendent of a classical academy in West Rutland. In November 1831, he was invited to become principal of a classical academy at Fort Covington in northern New York, where he also began to study law. In December, 1832, he returned to his father's residence in Castleton and continued his study of law while also teaching at the Castleton Academy.

In October 1834, Buell moved to Detroit, Michigan and within two months became Deputy Register of the Court of Probate. Because of his language skills, he was able to converse with the many French speakers native to Detroit at the time, as well as the increasing number of German immigrants. He was admitted to the bar in the spring of 1835 and commenced practice in Detroit.

Buel was city attorney in 1837 and a member of the Michigan State House of Representatives in 1838 and again in 1848, serving as speaker the latter year. He was the prosecuting attorney for Wayne County 1843-1846.

Buel first ran for the United States House of Representatives in Michigan's 1st congressional district in 1846, but lost in a close election. In 1848, Buel was elected as a Democrat from Michigan's 1st District to the United States Congress for the Thirty-first Congress, succeeding fellow Democrat Robert McClelland who did not run. Buel served from March 4, 1849 to March 3, 1851. He was an unsuccessful candidate for reelection in 1850 to the Thirty-second Congress and resumed the practice of law. He was again a member of the Michigan State House of Representatives in 1859 and 1860. He was appointed postmaster of Detroit on September 28, 1860, and served until March 18, 1861.

Buel married Mary Ann Ackley of West Rutland, Vermont, in October 1836, and they raised four daughters. Buel died in Detroit and is interred in Elmwood Cemetery there.