No, You Go First

Alice and Carol are best friends. Alice is having a fling with Carol's boyfriend Bob and wants to do the right thing and come clean to Carol. The conversation would usually go like this:

Alice: Carol, I have something to tell you.
Carol: Actually, I have something to say.
Alice: OK.
Carol: Bob, proposed to me! I'm so happy! This is the best thing that has ever happened to me! And I want you to be my maid of honour, because you have been such an amazing friend to me. Sorry, you were going to say something.
Alice: Oh, it's nothing.

In summary, a major confession is put on hold, when the other person says something that makes it even harder to confess. Usually the other person will find out from another, less appropriate source or situation.

Another common case is when Bob is about to declare his crush to Alice, and Alice (who sees him as a trusted friend) tells him first that she is in love with Charlie.

Examples of No, You Go First include:

Film

In Sky High, Layla is going to confess to Will that she likes him, by asking him to the school dance. As she was going to talk about the school dance, Will figures it's the perfect time to mention that he's going with the most popular girl in school. So Layla ends up telling him that she's going as well. With his archenemy.
This happens in Chicken Run between Ginger and Rocky. To elaborate; Rocky has decided to tell Ginger that he can't really fly like he told everyone he could. They meet on the roof of a hut, and Ginger thanks him for saving her life earlier in the movie. She then starts talking about how excited she is to finally be escaping the farm tomorrow (Rocky's supposed to give a flying demonstration for them.) After trying to break the news to her gently, Rocky gives up.
A variation is used in Old School (2003?), except one of the main characters is interrupted by one of the worst coincidences in the world.
Happens almost verbatim in How To Train Your Dragon:

Stoick: "Hiccup?"
Hiccup: "Dad! Uh, I have to talk to you, Dad."
Stoick: "I need to speak with you, too, son."
Stoick, Hiccup: [They both take deep breaths, then they both speak at once] "I've decided I don't want to fight dragons./I think it's time you learned to fight dragons."
Stoick, Hiccup: "What?"
Stoick: "Uh, you go first."
Hiccup: "Oh no, you go first."
Stoick: "All right... You get your wish: dragon training. You start in the morning."
Hiccup: "Oh, man, I should have gone first!"

In Drillbit Taylor, Drillbit starts to gear up to tell Lisa, a teacher he has fallen for, the truth about him (he's actually a homeless guy pretending to be a substitute teacher to protect three kids from a bully), but before he can get to that, she starts going on and on about how glad she is that he's such a great guy, since she usually dates absolute losers with nothing going for them, so he loses his nerve.
In Win A Date With Ted Hamilton, Rosalee and Pete both have something to tell each other. Pete was about to confess his love for her, but Rosalee goes first and tells him that Tad invited her to go back to Hollywood with him. At first he gives up, saying, "it was nothing," but then he plucks up his courage and kisses her on the lips.

Live Action TV

Every other episode between Chuck and Sarah.