Many quotes specially from Edward that melt the hearts of fans:
"If I could dream at all, it would be about you. And I'm not ashamed of it."
"For almost ninety years I've walked among my kind, and yours, all the time thinking I was complete in myself, not realizing what I was seeking. And not finding anything, because you weren't alive yet."
"Before you, Bella, my life was like a moonless night. Very dark, but there were stars points of light and reason and then you shot across my sky like a meteor. Suddenly everything was on fire; there was brilliancy, there was beauty. When you were gone, when the meteor had fallen over the horizon, everything went black. Nothing had changed, but my eyes were blinded by the light. I couldn't see the stars anymore. And there was no more reason for anything."
"You are the only one who has ever touched my heart. It will always be yours".
"Isabella Swan? I promise to love you forever, every single day of forever. Will you marry me?".
"Bachelor parties are designed for those who are sad to see the passing of their single days. I couldn't be more eager to have mine behind me. So there's really no point".
"I heard you."
"I'll always want you. Forever".
"I love you. I will always love you, no matter what happens now".
"Mostly I dream about being with you forever".
"Until your heart stops beating, Bella. I'll be here fighting".
"I'm just here to be your friend. Your best friend, one last time".
Jasper: (After telling the story of how he met Alice)
"I took her hand, and for the first time in almost a century, I felt hope."
(After Bella is worrying about James hurting anyone because of her) "I can feel what you are feeling now and you are worth it".
In New Moon, when Charlie phones Billy and has a go at him over Jacob 'betraying' Bella. .
And in Breaking Dawn, Edward's line to Jacob near the end "My brother...my son".
Charlie, an unconditional, generally unappreciated Papa Wolf.
Moreso in the movie, which eliminated the whole "Good for you, kid, forcing a kiss on my daughter, who now has a mysterious broken hand" bit.
The entirety of Alice and Jasper's relationship. Everything.
"More than my own life." Crosses over with Tear Jerker.
Jacob and Bella growing closer together in New Moon, as well as Jacob and Leah growing closer to one another in Breaking Dawn.
In Breaking Dawn Part 2, even though it was all a vision, Marcus, being the only one in the Voltori to accept his death, even muttering "Finally," is pretty heroic, considering most of the Voltori were running for their life.
More notably, during the envisioned battle, there is a scene when Esme is being pulled into Benjamin's chasm by an enemy vampire. Leah attacks him and pulls him off of her, saving Esme while dooming both herself and the attacker. Pulled up to Tear Jerker status when you see Leah's sad glance as she falls to her doom. Even more heartwarming when you consider that Leah never even *liked* the Cullens, but Esme had always been nice to her.
The ending sequence where Bella finally takes down her mental shield and shows Edward her memories of them together is an amazing way to end the series.