How was the first exoplanet discovered? What method was used to detect it?
This question is a bit tricky because it's uncertain as to what counts as "discovering" an exoplanet. Today, we count an exoplanet as having been discovered if it can be detected either by two separate techniques or three separate transits (if found via the transit method). Some of the very very early hunts for exoplanets did indeed find them, but many of their initial results were dubious or questionable (and sometimes wrong) and weren't really confirmed until much later. So do you count a planet as being discovered when it was first potentially detected or do you not count it until its been completely confirmed?
First Exoplanet Detection
The very first time someone could claim they discovered an exoplanet was in 1988. The astronomers Bruce Campbell, G. A. H. Walker, and Stephenson Yang announced that they had discovered a planet around the star Gamma Cephei using the radial velocity method. However, their results were not that great since, at the time, the detection was at the threshold of their technological capabilities and their results weren't entirely believable. It wasn't until years later that this planet was actually confirmed as existing. Whether you count this as the first real exoplanet discovery or not depends on what you define as a discovery, given that the actual confirmation didn't come until years after the first detection.
First Confirmed Exoplanet Detection
In 1992, Aleksander Wolszczan and Dale Frail managed to find and confirm a planet around a the pulsar PSR B1257+12, again with the radial velocity method. At this time, their measurements were good enough to be believed and so considered "confirmed".
First Confirmed Exoplanet Detection Around a Main-Sequence Star
This case is particularly called out because it was the first time an exoplanet was detected and confirmed to exist around a normal main-sequence star. This occurred in 1995 around the star 51 Pegasi. This discovery was made by Michel Mayor and Didier Queloz. Many people would consider this the first "true" exoplanet discovery and will cite this case if you ask when the first exoplanet discovery was made.
The first exoplanets to be discovered orbit about neutron stars, and pulsars in particular.
Pulsars that do not have planets and that are not a part of a multiple star system exhibit an extremely regular frequency. This frequency is modulated by the orbital frequency in binary systems that contain a pulsar. A pulsar with planets will exhibit a similar (but reduced) frequency modulation because the pulsar and planets orbit one another. The timing techniques used to determine whether pulsars are members of a binary systems enabled the discovery of the first exoplanets, in 1992.