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From ancient astrologers to modern man, eclipses are phenomena that capture the imagination. The mystery they invoke adds to their fascination. As a service to their kings, the oldest astrologers used them extensively to forecast droughts, famines, wars, and other major events; thus, eclipses became known to portend evils. Some called them omens—a decidedly final testimony, because they appear to operate outside the boundaries of our usual free will and control. But for all the great genius that went into the notes on eclipses handed down by the old mathematicians, there is little agreement among modern astrologers as to how long an eclipse is active. The formula passed along for measuring the life span of an eclipse was muddled or tampered with, leaving us with several different formulas to take into account. Without established basic criteria, assessing the impact of an eclipse is prone to inaccuracies and error.  - Astrologer Celeste Teal Eclipses: Predicting World Events and Personal Transformation.

About 4,000 years ago in China, the Chinese astronomers kept accurate records. We have the first actual documentation of an eclipse from 2134 [B.C.E.] In ancient times, in Chinese cultures and many other cultures around the world, eclipses were viewed as omens of doom. This was a negative message.

The Chinese, mentioned earlier, in 2000 B.C.E., as well as the Babylonian astronomers, kept very accurate records about the movements of celestial objects. The Babylonians, in the same general time frame, were able to come up with this 18-year Saros cycle that astronomers even use today to predict the recurrence of eclipses.
But, predictions have been somewhat crude, until the middle of the 1800s. Before that, astronomers could get a general sense of maybe what day an eclipse was going to occur, but didn't have much information about where you were going to be able to see a total solar eclipse. That all changed in the mid-1800s, when several things happened: There was a mathematician who came up with some equations that make it much easier to calculate the orbital motion of the moon and the sun, and the earth to be able to predict eclipses. And then another astronomer spent most of his adult lifetime calculating the paths for total solar eclipses, for a period of about 3,000 years. His name was Theodor von Oppolzer, and he published his canon of solar eclipses in the late-1800s, and with that change, astronomers — and the public as well, who had access to that information — were able to get a much better prediction of when and where a total solar eclipse was going to occur."

On impacts of ancient eclipses -

In 585 B.C., there were two Middle Eastern armies, the Lydians and the Medes. They were in battle one day, and unbeknownst to them in advance, they were right in the path of a total solar eclipse that day. And so, when the moon covered the sun and it went dark, the warring factions took note, and they laid down their arms and they made peace right on the spot.

On a famous eclipse mentioned in the Bible, which we can date -

It's from Amos, chapter eight, verse nine, that says, 'And on that day, says the Lord God, I will make the sun go down at noon and darken the Earth in broad daylight.' Based on other information about what was going on at the time, historians were able to go back in time, and they were able to fix the date as June 15, 763 B.C.E. And even today, as new historical records come to light that mention eclipses, it's kind of like a time marker that we have for over the millennia, and historians have found it very useful to go back and fix the dates and get a better idea of the chronology of certain events.

On Columbus' experience with eclipses -

On his fourth voyage to the Americas — and this was in 1504 — he got stranded in Jamaica, the natives fed him and his crew for a while, but after a few months, they got tired of it and withheld the food. Columbus knew from his navigational and astronomical tables that there was going to be a total lunar eclipse, and this was on Feb. 29, 1504. And so, shortly before the beginning of the eclipse, he called the natives together and said, 'If you don't resume feeding us, I'll make the moon disappear.' And they scoffed at him at first, and then he retired to his cabin, and sure enough, the moon started to disappear, and the natives relented. They fed him until he got rescued by other ships from Europe.

On how eclipses helped Einstein -

One of the things he postulated in his [theory of relativity] was that light would be bent by gravity. Back then, there was no way to test it on Earth, but they realized that during a total solar eclipse, the light coming from a star that was very near the apparent location of the sun in the sky would actually be bent by passing next to the sun, and you could measure its location during an eclipse. And then months later, when the moon was in a different part of the sky, they could measure the relative position of the star, and sure enough, they proved that the position changed and it was changed just according to what Einstein calculated. So, it was the first empirical proof of the theory of relativity.*

More:

Many authorities believe Stonehenge was constructed for the purpose of recording and monitoring eclipses. More on how to predict eclipses via Stonehenge can be found here http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0102-1-stonehenge/eclipses.htm

Ancient civilizations explained eclipses as evil spirits attacking the Sun or Moon. It was customary to run into the street, shout, scream and make as much noise as possible to frighten the spirits away.

Some thought a huge black dragon swallowed up the Sun or Moon. The places in the zodiac where eclipses occur are known as 'the Dragon's Head' and 'the Dragon's Tail'.
History records numerous reports of fearful Roman emperors trying to deflect the personal effects of an eclipse by murdering their top ranking statesmen on the advice of astrologers.

The prospect of the eclipse on 'Black Monday', 29 March 1652, had the citizens of London so terrified that it was reported: "hardly any would work, none would stir out of their houses.... The rich fled London, the poor prayed for deliverance and a thriving trade was conducted in cordials claiming to allay the effects of the eclipse". To the embarrassment of astrologers, the eclipse passed without any notable disaster. This solar eclipse that blotted out the sun in Scotland and Ireland was widely interpreted as the beginning of God’s wrath, a sign of the imminence of the Day of Judgment.

* http://www.wbur.org/hereandnow/2017/08/15/solar-eclipse-history