By looking on a distant star, let us call that star for Star A. Star A is far away from another star called Star B, but is aligned behind it when observed by telescope from earth. It will lead to a deformation of light in to two distorted images of Star A. If there is an planet orbiting Star B it produces a third image of Star A.
When light from Star A pass on all sides of Star B it produces a ring of light around the object and that phenomena is known as an Einstein ring.
Another method of detecting exoplanets is to measure the magnitude of the star over time. When an exoplanet pass in front of the star there will be a small drop in brightness. This drop will occur for every revolution of the planet. The decrease is very small depending on the planet mass, usually between 0.01 percent and 1 percent.
As the star's mass and size can be determined from spectroscopic analys of the observation then also the mass and the distance of the exoplanet can determined.
Radial velocity method is also known as Doppler spectroscopy or the wobble method. This measurement is done by recording variations in the color of light from the star. Even for a smaller object like an exoplanet orbiting a star, the gravitational influence of that planet can cause the star to move in a tiny circle. That because the star and the planet orbiting a common center of mass.
Light is electromagnetic radiation and has a wavelength, the visible spectrum is between blue light 380 nm and red light about 740 nm. A common example of the doppler effect is when a ambulance is approaching an observer its siren sound higher in pitch and when it is receding it sound lower in pith. That is because the wavelength of the sound is shorter and has a higher frequency when the ambulance approaching and the wavelength is longer and has a lower frequency when it is receding. Same physics law applies when a star moves towards the earth the wavelength is then shorter and the light from the star color spectrum will shift towards blue. When a star moves from the earth the wavelength is longer and the light will shift towards red color.
A pulsar is an neutron star, a very heavy dense and small object that remains after a supernova explosion. A pulsar emit radio waves when they rotate. If a pulsar has an exoplanet it will behave just like a regular star it will move in its own orbit around a common center of mass. Therefore it could be detected comparing the changes of the radio waves wavelengths. With this method it is possible to detect very small exoplanets. Unfortunately these exoplanets won't be habitable because of the intensity of the high energy radiation from the neutron star
This methods means that the exoplanets has been detected directly from telescope image. Exoplanets has a very faint light compering to the host star which makes this a very difficult method. As the planets reflect very little star light they are detected studying their thermal emission instead. It is easier to detect a planet if it is very large and orbiting close to their star so called hot jovians.