A Force-sensitive was a being who was highly attuned to the flow of the Force. Force-sensitives could, with training, learn to sense and manipulate the Force. The Jedi of the Old Republic took Force-sensitive children at a young age to become a part of their Order.
Trained Jedi could sense high levels of the Force in persons. The presence of those persons in a vicinity could also be sensed. For more accuracy, they performed midi-chlorian counts to detect Force-sensitivity in candidates. There were also several other testing technologies developed to hunt Force-sensitives during the Great Jedi Purge. Species with high number of Force-sensitives included Humans and Ithorians. Species producing low number of Force-sensitives included the Wookiees.
Qui-Gon Jinn discovered Anakin Skywalker through his prowess at Podracing. Anakin himself once believed as a boy that he had a 'string of lucky guesses', when actually these 'guesses' were early indicators of the Force. His mother Shmi Skywalker noticed these anomalies in her son as well, and described Anakin Skywalker as 'special'. Indeed he was, for Skywalker had no biological father; Shmi was a virgin mother..
Other potential Jedi have been discovered in roughly the same way; when there seemed to be an unusual amount of luck associated with an individual.