Before the seventeenth century, it was generally thought that light is transmitted instantaneously. This was supported by the observation that there is no noticeable lag in the position of the Earth's shadow on the Moon during a lunar eclipse, which would otherwise be expected if c were finite. Nowadays, we know that light moves just too quickly for the lag to be noticeable. Galileo doubted that light's speed is infinite, and he devised an experiment to measure that speed by manually covering and uncovering lanterns that were spaced a few miles apart. We don't know if he ever attempted the experiment, but again c is too high for such a method to give an even remotely accurate answer.
The first successful measurement of c was made by Olaus Roemer in 1676. He noticed that depending on the Earth–Sun–Jupiter geometry, there could be a difference of up to 1000 seconds between the predicted times of the eclipses of Jupiter's moons, and the actual times that these eclipses were observed. He correctly surmised that this is due to the varying length of time it takes for light to travel from Jupiter to Earth as the distance between these two planets varies. He obtained a value of c equivalent to 214,000 km/s, which was very approximate because planetary distances were not accurately known at that time.
In 1728 James Bradley made another estimate by observing stellar aberration, being the apparent displacement of stars due to the motion of the Earth around the Sun. He observed a star in Draco and found that its apparent position changed throughout the year. All stellar positions are affected equally in this way. (This distinguishes stellar aberration from parallax, which is greater for nearby stars than it is for distant stars.) To understand aberration, a useful analogy is to imagine the effect of your motion on the angle at which rain falls past you, as you run through it. If you stand still in the rain when there is no wind, it falls vertically on your head. If you run through the rain, it comes at you at an angle and hits you on the front. Bradley measured this angle for starlight, and knowing the speed of the Earth around the Sun, he found a value for the speed of light of 301,000 km/s.
The first measurement of c that didn't make use of the heavens was by Armand Fizeau in 1849. He used a beam of light reflected from a mirror 8 km away. The beam was aimed at the teeth of a rapidly spinning wheel. The speed of the wheel was increased until its motion was such that the light's two-way passage coincided with a movement of the wheel's circumference by one tooth. This gave a value for c of 315,000 km/s. Leon Foucault improved on this result a year later using rotating mirrors, which gave the much more accurate value of 298,000 km/s. His technique was good enough to confirm that light travels slower in water than in air.