It's not surprising that it took so long. The stars are so far away that even the light from our hundred nearest neighbors takes up to twenty years to get here. In addition, we have to remember that we see Solar System planets by reflected sunlight. If we were 24 trillion miles away - the distance to the next star - even Jupiter would be submerged in the Sun's light.
This means astronomers don't find extrasolar planets simply by pointing their telescopes at nearby stars. In fact, it was only in 2005 that we had the first image of an extrasolar planet. The technology for imaging has improved to the point that some three dozen extrasolar planets have been discovered by direct imaging. However that isn't many out of nearly two thousand.
So how do they find the planets if they can't see them? Usually by detecting a planet's influence on its star.
The method which had enabled the discovery of most of the extrasolar planets in the first fifteen years of discoveries is Doppler spectroscopy. It's also called the radial velocity method or, popularly, the wobble method.
Think how the note of an emergency vehicle's siren changes as it approaches and then passes you. When approaching, the vehicle overtakes the sound waves. This increases the frequency you hear, making the note higher. After it passes, the opposite occurs and the pitch drops. This is
The Doppler effect applies to light waves too. The spectrum of approaching objects is blue-shifted, i.e., it's detected at a higher frequency. Objects moving away are red-shifted.
Strictly speaking, planets don't orbit stars. There is a mutual gravitational attraction so that star and planet orbit their common center of gravity. This center is inside the star, so the interaction makes the star wobble slightly. If the orbiting planet causes the star to move alternately towards and away from us, a sensitive telescope may detect alternating blue and red shifts in the light spectrum. The frequency of the shifts shows the planet's orbital period and the size of the shifts tells us about the mass.
In 1995 Swiss astronomers Michel Mayor and Didier Queloz discovered the first extrasolar planet orbiting a sun-like star. It was quite a surprise, for it had at least half the mass of Jupiter, but was in an closer orbit than Mercury's is to the Sun. This was the first of the "hot Jupiters".
Hot Jupiters aren't the most common planets in the Galaxy, but they were the easiest to find. Massive and close to the star, their gravitational influence is maximized. And it doesn't take long for repeated observations to establish the orbital time. By contrast, Jupiter itself takes twelve years to orbit the Sun.
However in 2010 half the discoveries were made by a different method - the transit method. A transit occurs when a planet crosses the disc of its star. This causes a tiny dip in the star's brightness. The size of the dip provides evidence of the planet's diameter, and the orbital period is determined from the timing of the dips. As there are many causes for variation in starlight, transits are confirmed by various follow-up methods.
Ever since 2011 most of the discoveries have been through the transit method and most of these were made from data collected by the Kepler mission. In January 2015 the 1000th extrasolar planet discovery by Kepler was confirmed.
NASA's Kepler mission, launched in March 2009, monitored a small, populous star field. Not being subject to distortion by Earth's atmosphere, its sensitive photometer was able to detect the transits of smaller planets. One of the mission goals was to find Earth-sized planets, and there are some possibles.
Kepler's main mission ended in 2013 when a second stabilizer failed, leaving it unable to carry out precision targeting. However the vast amount of data collected is still being analyzed, and new discoveries continue to be made. In addition, in 2014 an ingenious solution was found that has allowed Kepler to start a new mission.
There are images related to this article on my Pinterest board Extrasolar Planets.