An affair was a romantic relationship outside the bonds of marriage.
In the 21st century novel Hotel Royale, Rita and a bellboy had an affair which ended in both being killed by Mickey D. (TNG: "The Royale")
Despite Trills considering them to be shameful, Curzon Dax had an affair with Enina Tandro in 2339. (DS9: "Dax")
In 2362, Marla E. Finn and William Hodges's affair was discovered by an enraged Walter Pierce, who killed them both before committing suicide at the Utopia Planitia Fleet Yards. (TNG: "Eye of the Beholder")
In 2365, Vaatrik Pallra believed her husband to have had an affair with Kira Nerys, although this was not the case. (DS9: "Necessary Evil")
A deleted scene from the episode suggested that Vaatrik Pallra had an affair with Dukat herself.
Nidell projected Fenna as a way to escape her marriage with Gideon Seyetik without cheating physically. (DS9: "Second Sight")
When Beverly Crusher, under the influence of Ronin, left the USS Enterprise-D in 2370, Jean-Luc Picard questioned her motives, telling her "this is more than just an obsesssive love affair". (TNG: "Sub Rosa")
According to Jadzia Dax in 2373, Klingon mating rituals were very "involved" and not about a one-night affair. (DS9: "Looking for par'Mach in All the Wrong Places")
When B'Elanna Torres relived Korenna Mirell's youth memories of her relationship with Dathan Alaris the same year, B'Elanna described it as more than a simple love affair. (VOY: "Remember")
In 2374, Brunt called the relationship between Zek and Ishka a "perverted little love affair". (DS9: "Ferengi Love Songs")