Born in Bolton, Massachusetts, Robinson was a catcher in the minor New England League in 1885 and made it to the major leagues in 1886 with the Philadelphia Athletics of the American Association, where he remained until 1890. He lasted in the majors until 1902, playing much of his career with two separate Baltimore Orioles franchises - from 1890-99 with the Orioles team which folded after the 1899 National League season, and in 1901-02 with the American League team which moved to New York in 1903 and became the Yankees. He also spent one season, 1900, with the St. Louis Cardinals.

Over the course of his career, Robinson played 1,316 games as a catcher, which prepared him for his second baseball career as a manager. The star catcher of the Orioles dynasty which won three straight titles from 1894-96, he compiled a career batting average of .273, with a peak of .353 in the heavy-hitting season of 1894. Durable behind the plate, he caught a triple-header in 1896, followed by a double-header the following day. He also was the first catcher to play directly behind the batter at all times, as the previous practice had been to play farther back when there were fewer than two strikes. A highlight of his career was a seven-hit game June 10, 1892, still the major league record (Rennie Stennett tied it in 1975).

Robinson served as player-manager of the AL Orioles in 1902, after his friend and former teammate John McGraw had departed to the New York Giants. Afterward, McGraw enticed Robinson to be his pitching coach from 1903 to 1913, during which time the Giants won five NL pennants. Robinson would not don the manager's cap again in the majors until 1914, when he took over the Brooklyn, New York franchise in the National League. The team was known by various nicknames, including Bridegrooms, Superbas, and Dodgers, but during Robinson's managerial tenure, which lasted until 1931, the club was as often referred to as the "Robins" in honor of their manager, who had acquired the nickname "Uncle Robbie".

In his 18 years at the helm of the Brooks, Robinson compiled a record of 1375-1341, including National League championships in 1916 and 1920 - Brooklyn's only pennants between 1901 and 1940. Those two clubs lost in the World Series to the Boston Red Sox and Cleveland Indians, respectively. His 1375 NL victories were, at the time, the 3rd-highest total in NL history, trailing only the totals of McGraw (then with 2652) and Fred Clarke (1602).

Robinson was highly regarded for his ability to draw outstanding performances from his pitching staffs, a result of his many years as a catcher. Among the pitchers he guided to success were Joe McGinnity with both Orioles teams and the Giants, Rube Marquard with the Giants, and Dazzy Vance and Burleigh Grimes with the Dodgers. Another pitcher who would later recall Robinson's excellent advice, although they never played together during a regular season, was John Tener, who in the 1910s served simultaneously as NL president and Governor of Pennsylvania.

On March 13, 1915, at spring training in Daytona Beach, Florida, Robinson decided to try to set a record of sorts by catching a baseball dropped from an airplane being flown 525 feet (160 m) overhead. Ruth Law, the aviatrix, supposedly forgot to bring a baseball and instead dropped a grapefruit, which splattered all over the manager. Robinson thought he was covered with blood until his teammates burst out in laughter. Outfielder Casey Stengel, later a successful manager himself, is generally believed to have convinced Law to make the switch.

After his retirement from managing, Robinson became the president of the Atlanta Crackers minor league team. He died in Atlanta, Georgia at 71 years of age following a brain hemorrhage, and was buried in Bonnie Brae Cemetery in Baltimore.