How did we realize that light travels at a finite speed? We all know that light travels at a finite speed of nearly $3\times10^8\mathrm{m\,s^{-1}}$. But how did we come to think that light has a finite speed? Why did the scientists try to calculate the speed of light? Or did we just shoot in the dark, thinking that maybe light has a finite speed, let us try finding its value?
 A: As far as I know the first time anyone proved that light has a finite speed was when the astronomer Rømer discovered variations in the timings in the transits of Jupiter's moons. He correctly attributed this to the time light took to reach Earth from Jupiter.
His calculated value for $c$ was about 26% too low, but that was pretty good given the state of the art at that time.
A: It was discovered through the Michelson-Morley Experiment. in 1887.
The main goal of that experiment was the discovery of aether, the medium through which light travels, as scientists hypothesized back then.
In their setup they had two light sources moving at perpendicular directions, with the whole experimental apparatus (light source, some mirrors, detector) moving at the same direction with one of the beams. What was realized, was that no matter what the speed of the apparatus was, or its direction, the experimental output was always the same, even when the apparatus wasn't moving at all.
A: Measuring light speed is rather trivial now a days, we have laser pulses which have durations of the order of 10^-14 sec, photodiodes which can measure the light pulses as fast as 10^-11 sec, and oscilloscope which can display these events with a resolution of 10^-9 sec. so overall you can detect a nanosecond (10^-9 sec) very easily. now what you can do if you have access to some laser facility you just take a fast laser pulse, split it into two parts steer one pulse by a known distance (lets say 1 meter) then make them fall on photodiode, connect the photodiode with a oscilloscope and you will see pulses on the oscilloscope and hence with a very little effort you can get the speed of light (if you have correct tools). 
This question is of historical importance now, maybe in some 1800 AD this would be really plausible question but there is no doubt that light has finite speed as of today. The only question is how accurately we can measure it.
Edit: I am sorry, I have misunderstood your question. It was correctly told by John Rennie that the astronomer Roemer was first to realize that light has finite speed. Roemer observed that the duration of eclipse of one of the moons of Jupiter is shorter when the earth is moving towards the Jupiter and longer when earth is moving away from Jupiter.
This is one of accidental discoveries, One such interesting discovery I remembered was presence of atmosphere on Venus. When Venus pass over the surface of sun (and world was well equipped to see it for first time in 1761) a white ring appeared around the Venus, and an Russian scientist Lomonosov has commented that Venus should have a atmosphere. 
