Resse is a pairing of Re/becca and J/esse. It is a marriage relationship. This is the first real steady relationship Jesse has had throughout the show.
In "Tanner vs. Gibbler", Jesse and Rebecca first met and Jesse quickly had feelings for her, but Rebecca wasn't interested yet. One reason she liked Jesse was because he reminded her of a young relative she has who does Elvis impressions, which Jesse act similar too and has done those impressions in "Mad Money".
In "A Little Romance", when Jesse is being bind in the auction by another women, Becky gets jealous and outbids that lady, even though Jesse has given the opportunity to go on a date with him before. This then becomes the day they begin their relationship.
In "Little Shop of Sweaters", Jesse and Becky go on a romantic date and they exchange gifts. Becky receives a new pair of earrings and Jesse gets an Elvis spoon. Right after Jesse tells Becky that he wants their relationship to be serious, Becky reveals that she has another date. Jesse is upset at this first, but Rebecca tells him that she's not ready for it to be serious. They end up kissing at the end.
In "Baby Love", they start thinking of their future children's names, even though they started to date. In "I'm There for You, Babe", Becky asks Jesse to come to her aunt's former son-in-law's wedding, but he is too busy. Then she and the rest of Jesse's family and friend help him during a gig at the Smash club and fail to entertain the audience well. After that, Becky helps Jesse feel better from what happened.
In "Luck Be a Lady (Part 1)", The family goes to Lake Tahoe and while Danny and Becky interview Todd Masters, Jesse gets jealous of Becky being close to Todd. This jealousy ends up with a misunderstanding and an argument with both of them. At the end of the episode, the couple makes up and they decide to get married.
In "Luck Be a Lady (Part 2)", right after what happened the episode before, Jesse convinces Becky to elope and to keep this a secret from the others. During the wedding, when they say their "I do"s, Becky immediately leaves the alter and a broken-heartened Jesse. Back at San Fransisco, Becky talks to Jesse and tells him that when they get married, they have to talk about it and it while have to happened not as soon.