Rømer used an Astronomical method.
Hippolyte Fizeau in 1849 used an earthbound method to measure the speed of light. He used a beam of light reflected from a mirror 8 km away. The beam was passed through the gaps between the teeth of a rapidly rotating wheel. The speed of the wheel was increased until the beam of light passed though the next hole and could be seen. Then c was calculated to be 315,000 km/s. Leon Foucault improved on this a year later by using rotating mirrors and got a more accurate answer of 298,000.
Hippolyte Fizeau’s experiment (http://schools-wikipedia.org/images/203/20343.png.htm)
Abraham Michelson was teaching at the time when he realised that he could improve on Fizeau’s experiment. He used 2,000 feet instead of 60 feet measured to one tenth of an inch. He used high quality lenses and mirrors to reflect the beam. His result was 186,355 miles per second; his margin of error was 30 miles per second. This experiment was more accurate than Foucault and Fizeau’s experiments. Michelson’s measurement of the speed of light was the most accurate measurement for the next 40 years; Michelson measured the speed of light again then.
Lois Essen and A.C Gordon-Smith in 1946 used a microwave cavity with precise measurements to make normal microwaves. As the wavelength was know from gemoetry of the cavity and from electromagnetic theory a calculation could be made to know the speed of light. Their result was 299,792 miles per second. When they had repeated it many times they came up with a new figure 299,792.5 miles per second. It became the value used by the 12th General Assembly of the Radio Scientific Union in 1957.
As this table shows as time went on, measurements of the speed of light got better and more accurate and precise. This was due to technological advances. The uncertainty of error got les and less.
Bibliography
http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html
http://schools-wikipedia.org/wp/s/Speed_of_light.htm
Britannica Concise Encyclopaedia:
Publisher: Encyclopaedia Britannica (UK) Ltd; New edition (7 April 2003)
By Dale H. Hoiberg (Editor), Theodore N. Pappas (Editor), Marsha Mackenzie (Editor)
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html- Original by Philip Gibbs 1997