The film is particularly notable for the intensity of the initial 20 minutes or so which depicts the Omaha beachhead assault.
warning: Wikipedia contains spoilers
The general plot of the film, as the title suggests, is essentially a humanitarian rescue mission led by John Miller, an army captain, played by Tom Hanks to return the last surviving Ryan brother from the Normandy front line to his mother. The movie also stars Tom Sizemore[?], Edward Burns[?] and Matt Damon. Many critics commented that the film seemed marred somewhat by Spielberg's propensity for sentimentalism.
Miller, as played by Hanks, conceals his erstwhile profession of schoolmaster and his background from the troops under his command; the uncovery of Miller's background becomes a sub-plot of the film in as much as the men have a pool on his origins, which he steadfastly refuses to reveal. Miller moreover is of a nervous disposition and this psychological leakage is explicitly communicated by his unsteady hands.
The bond between Miller and his men is forged in the beachhead assault on a German bunker, where his decisive action saved the day.
As the position consolidates, Miller is given his new assigment, to find Private Ryan, who had been parachuted in as a member of the 101st Airborne, which, as the film historically correctly asserts, was scattered widely across Normandy. Ryan is the sole surviving member of four brothers, the other three having been killed in action. The American command takes the decision to bring him back for his mother's sake.
Eventually, at the expense of a number of members of their unit, Miller and his men catch up with Ryan. They break the news of his brothers' deaths to him and tell him that he is going home. Ryan is torn in his decision but elects not to desert his strategically important post. Miller and his men protect him, and all but a few and Ryan survive a ferocious German tank assault on the bridge which they are defending. Miller is killed in the assault.