Relationship is one of the most key elements in improv. Unlike those rules that you were told were so important: yes and, listening, staying away from questions, etc, relationships are not key to every scene, yet the relationship between characters is arguably the most powerful tool used to start a scene and keep it alive and well.
For example, let's say you had any plain old scene start, we'll pick two guys working at a factory. They are just average Joes, working the levers and cranks from 9 to 5. What they are doing is not truly enthralling, which perhaps lends credence to someone introducing an action or a conflict and raising the stakes. However, instead of making the scene about a certain object or activity, we first can focus on these people and their relationship with each other.
Maybe one of them, Dave, used to go to school with the other, we'll just call him George. Dave and George even were on the swim team together, except Dave wasn't particularly skilled in the breastroke. On some fateful day, Dave cramped up and had to be rescued by George in a heroic manner, making them absolutely inseperable, though their manly pride doesn't allow them to show it.
From here, the scene could go in a few different directions. It certainly opens up an opportunity for George to save Dave's life again, or maybe Dave will finally be able to repay the favor when the Twine Reel goes haywire as George is cleaning it. Or maybe we continue to focus on the two of them, until a third party enters and heightens the action by introducing some conflict that we know goes against the grain of everything Dave and George have been through. They leave work and a girl that has always been the point of contestation between the two shows up? The two happen to run into an Olympic champion swimmer who can take on one and only one protégé? The factory boss has to let one of them go, but he has a change of heart when faced with how devoted Dave and George are to each other?
There are many possibilities that open up when we fully explore who are characters are, what they want, and what has happened between them. To know who you are playing on stage and how he/she relates to the other characters is priceless, because it allows you build believable scenework using yesand that is focused on the characters themselves. And, of course, characters on stage are ultimately more interesting than an imaginary object or a vague threat that seems hollow and two-dimensional. Good improv works beyond zany spontaneity to make any scene seem realistic, no matter what is happening, and knowing how to express relationships between characters is a staple of any improviser's technique.