In a 1969 article in BCM Sir Richard Clarke, who developed the British Chess Federation rating system, notes that Brumfitt was the first person to suggest - also in the British Chess Magazine, 1891 - the need for a rating system. But he also mentions that any real attempt to devise one was sporatic and isoated until after WWII.

In 1948 Anton Hoesslinger (1875-1959) introduced the "Ingo system" - named for his hometown Ingolstadt, Bavaria. He worked as a postal supervisor, a job that found him traveling to various places where he also managed to play chess. Since, as a stranger, he was always treated as an unknown quantity, he saw the need for a system that could somehow define his chess strength. The system he devised formed a basis for later systems and was very popular in Germany for a long time.

Some later systems were the Harkness system, developed in the 1950's for the USCF by Ken Harkness (who also developed the swiss system for tournaments). The Harkness system evaluates the strength of a player's opposition by summing the final scores of his or her opponents and then disccarding the highest and lowest of these scores and because of this, it's sometimes called the Median system. A modification of this system, the Solkoff system, uses a similar formulae but no opponents' scores are discarded. Both these systems sometimes lead to inaccuracies. For this reason the most commonly used system today is the ELO system. Other systems used today are the aforementioed  BF system by Richard Clarke and the Glicko system  by Professor Mark E. Glickman of Boston University. This compicated system is explained in detail here and here.

I'm not a mathematician and discussions of ratings systems make my mind go blank, but Arpad Elo is a truly facinating man even without the ELO fame.
 

The ELO system is named for Arpad Emrick Elo, born Arpad Emre El? August 25, 1903 in western Hungary near the small beet-producing town of PApa. His parents were farmers. They emigrated to the U.S. in 1913 and settled in Cleveland, Ohio. There Arpad discovered chess and taught himself the game. Chess was just a hobby along with wine-making, horticulture, astronomy and music. Arpad attended the University of Chicago, receiving a B.S. and M.S. in physics. From 1926 until 1969, he taught physics at Marquette University, a Catholic university founded by the Jesuits in 1881 in Milwaukee. Later, he also taught at the University of Wisconsin, also in Milwaukee on a part time basis.

Although chess was just one his many passions, he excelled in the game, winning the Wisconsin state championship a total of eight times. He drew two games against Reuben Fine, one of the world's best players.

Chess in Milwaukee was played under the auspices of the Western Chess Association. In 1935, the Western Chess Association became American Chess Federation and Elo was elected president (it's said he was late for that particular meeting, involved in one of the aforementioned games with Fine that resulted in a draw, and found when he arrived at the meeting that he was elected president in absentia). He was president from 1935 til 1937. In 1939, the American Chess Federation joined with the National Chess Federation to form the United States Chess Federation (USCF). Elo was a charter member.

in 1959, Elo was approached by the USCF president, Jerry Spann, about coming up with a more reliable system (than the Harkness system which was in place) of calculating ratings. The USCF adotped the ELO system in 1960 and by FIDE in 1970. Until 1980 he did all the rating calculations for FIDE, first with a paper and pencil, then, when the calculator was invented, he used a Hewlett-Packard calculator.

In 1978  he wrote a book called, The Rating of Chessplayers Past and Present. Edited by fellow Milwaukeean, Fred Cramer, this book examined the historical ratings of 476 chess players and did statistical analysis using various demographic variables such as players' ages when the learned to play and certain genetic aspects. He extrapolated his data to come up with a rankng list of players throughout history. Here's the top 20 (excluding Fischer and Karpov both of whom topped the list with 2780 and 2775 respectively, but were dropped from the January 1, 1978 FIDE ratings list, since these numbers did not represent a five-year average for the players):

Jeff Sonas does something similar at chessmetrics.

In 1988, Arpad Elo was inducted into the Chess Hall of Fame.

Elo's son, Arpad Elo, jr., born on December 24, 1925, is a chemist holding 2 patents, a musical director, pianist, clarinetist and basset horn player, composer of 80 pieces for orchestra, band, chorus, solo voice, keyboard, and various chamber combinations and a 18th century instrument reconstructionist.

Like father, like son.