Astronomers have discovered exoplanets using a number of different methods:
Radial Velocity Method
The vast majority of exoplanets were discovered through the gravity force they exert on their parent star. In other words they make their stars 'wobble' about as the star-planet system circles around a common centre-of-mass.
If we look along the plane of the planet's orbit we see the wobble as movement of the star towards and away from us. If we are looking down on the system from above, as in the animation, then we see the wobble as an astrometric shift, i.e. the star does a small circle in the sky when compared to other nearby stars which are fixed in position.
However, astronomers find it much easier to spot motion toward and away from us, the star's radial velocity, by observing its spectrum. Current technology allows us to detect radial velocities of just 1 metre per second - a fast walking pace. Jupiter causes the Sun to wobble by up to 12.5 metres per second, so it is no surprise that astronomers are now finding Jupiter-like planets.
Another method of finding exoplanets is simple enough that we can do it with the Liverpool Telescope. The method works by looking for regular dips in the brightness of a star as a large planet passes, or transits, in front of it
For close-orbiting giant planets, this dip in brightness can be as much as 2% of the original brightness and occurs every few days. In the wobble animation (above), the planet is orbiting at right-angles to us. As a result, there would be no chance of it passing in front of the star, but we now know of dozens of cases where transits do occur.
In fact, given the millions of stars in our galaxy, the chances are that many more are inclined such that a planet will block out a fraction of the parent star's light at some point. It is thought that around 1% of stars might have a transiting planet, although it is difficult to catch one in the act.
Direct Imaging Method
This technique aims to get an actual image of the planet as illuminated by the light from its parent star. This is more difficult than it may seem.
The star can be thousands of times brighter than the planet, such that the planet image is lost in the glare. Imagine trying to spot a candle in a searchlight. Also, the vast distances involved mean that telescopes find it hard to resolve (or separate) the star and planet, especially if they are close-orbiting planets.
Astronomers have developed instrumentation that reduces the contrast between the planet and star, such as using a "coronagraph" - a physical mask to block out light from the star. They also use techniques that reduce the effects of atmospheric turbulence, like using a space-based platform.
The image here was taken in 2005 and shows a planet (b) orbiting the star GQ Lupi (A) at a distance of twenty times the orbital distance of Jupiter.