The line, "Think about the poor bastard's mother," when they're all bitching about why they have to rescue Ryan.
The whole D-Day landing scene at the beginning. Often imitated, never duplicated, entirely tear-wrenching at the seeming futility of it all and the incredible, undeniable vision of death and hell on a beach.
To be followed up by a series of brilliant scenes where Mrs. Ryan finally gets all three telegrams on the same day. The imagery was just perfect - from the ceaseless typing of telegrams to the four stars on Mrs. Ryan's window. One for each son, three of whom were never coming home. One of the most powerful scenes in the film, moreso because it has no dialogue.
But, in particular, the completely wordless scene in which Mrs. Ryan watches the car driving up the road, goes out to see what it's about, looks slightly uneasy when a senior army officer climbs out, but when he's followed by a priest, her legs just give way under her. Great acting, great cinema.
For some reason, scenes with people collapsing because of shock and grief always gets me. Always.
When General Marshall says "...that you are the mother of five sons who have died gloriously on the field of battle."
That's actually a quote from a letter Abraham Lincoln wrote to a mother of five during the American Civil War, and it is the impetus for Marshall's orders to bring James Ryan home.
Doc Wade's death. The sound of his crying out for his momma as he bleeds out is about as heartbreaking as it gets.
Particularly since you can spot exactly when he realizes that he's going to die, as well as the other soldiers' facial expressions when they figure it out too ("...m-more... morphine...") and him saying he wants to go home & crying for his mama as he dies. Combine that with his earlier monologue about his mother, and you have a brilliant, but damn depressing scene.
While the first Ryan they find turns out to be the wrong James Ryan, the way he reacts to the news, bursting into tears and begging to go home, is heart-breaking.
Private Ryan refusing to go home because he feels that the men he's serving with deserve it just as much as him, that he's not going to abandon them and they're the only brothers he has left.
Ryan's story about his brothers always does it for me. As funny as the story is, throughout the telling of it, Ryan is visibly breaking up about the knowledge that all of his brothers are dead.