Jessie's backstory must have a direct mainline to your worst fears of abandonment and lost childhood for all the tears it extracts. Sarah McLachlan singing "When She Loved Me" in the background gives it a feeling of true love lost, too.
The leadup where Jessie reveals she knows exactly what Andy means to Woody, and why:
Woody: Well if you knew him you'd understand Andy is a-
Jessie: [bitter] Let me guess, Andy is a real special kid and to him you're his buddy; his best friend and when Andy plays with you it's like... [wistful] even though you're not moving, you feel like you're alive, because that's how he sees you.
Woody: How did you know that?
Jessie: Because... Emily was just the same, she was my whole world.
What Jessie says immediately after finishing the song.
Jessie: You never forget kids like Emily or Andy... but they forget you.
Woody:(heartbroken) Jesse, I didn't know.
Jessie: Just go.
The DVD commentary stated that Joan Cusack, Jessie's VA, also cried while recording the last line.
Tom Hanks, in an interview on the radio show Wait, Wait, Don't Tell Me, revealed that when he and Tim Allen (Buzz) were watching an early release of the movie, after this scene, they both looked at each other across an empty theater, tears in their eyes.
This quote from John Lasseter from The Pixar Story brilliantly describes this scene:
"At that moment you know that no one's thinking 'Well this is just a cartoon', 'this is just a bunch of pencil drawings on paper' or 'This is just a bunch of computer data'. No. These characters are alive and they're real."
The scene where Woody refuses to go back with Buzz and the rest of the gang before the scene below was also tear jerking, as well.
Woody: I don't have a choice, Buzz. This is my only chance.
Buzz: To do what, Woody? Watch kids from behind glass and never be loved again? Some life...
The scene where Woody realizes he needs to go back to Andy after seeing an old TV clip of himself singing "You've Got a Friend in Me" in the most straightforward and sweetest way if that makes any sense.
"I can't stop Andy growing up. But I wouldn't miss it for the world." It gets more poignant given the events of the third movie.
When Stinky Pete gives his backstory on why he's so adamant to go to Japan, even if it's by force.
Jessie: Prospector, this isn't fair!
Stinky Pete: Fair?! I'll tell you what's not fair! Spending a lifetime on a dimestore shelf, watching every other toy be sold! Well, finally my waiting has paid off, and no hand-me-down cowboy doll is gonna MESS IT UP FOR ME NOW!
When Woody looks up, after Andy has done his own patch job on Woody's torn arm, with a sweet smile on his face, knowing that Andy cares for him all along...