Marko Alexandrovich Ramius, a Lithuanian submarine commander in the Soviet Navy, intends to defect to the United States with his officers on board the experimental nuclear submarine Red October, a Typhoon-class vessel equipped with a revolutionary stealth propulsion system that makes audio detection by sonar extremely difficult. The result is a strategic weapon platform that is capable of sneaking its way into American waters and launching nuclear missiles with little or no warning.
The strategic value of Red October was not lost upon Ramius, but other factors have spurred his decision to defect. His wife, Natalia, died at the hands of a doctor who was incompetent and intoxicated; however, the doctor escaped punishment due to his status as the son of a Politburo member. Natalia's untimely death, combined with Ramius' long-standing dissatisfaction with the callousness of Soviet rule and his fear of Red October's destabilizing effect on world affairs, exhausts his tolerance for the failings of the Soviet system.
As the ship leaves the shipyard at Polyarny, Ramius kills Ivan Putin, his political officer, to ensure that Putin will not interfere with the defection. Before sailing, Ramius had sent a letter to Admiral Yuri Padorin, Natalia's uncle, brazenly stating his intention to defect. The Soviet Northern Fleet therefore sails out to sink Red October under the pretext of a search and rescue mission. Meanwhile, Jack Ryan, a high-level CIA analyst and an ex-Marine, flies from London to Langley, Virginia, to deliver MI6's photographs of Red October to the Deputy Director of Intelligence. Ryan consults a friend at the U.S. Naval Academy, ex-submariner Skip Tyler, and finds out that Red October's new construction variations house its stealth drive.
Red October passes near USS Dallas, a Los Angeles class submarine under the command of Cdr. Bart Mancuso, which is patrolling the entrance of a route used by Soviet submarines in the Reykjanes Ridge off Iceland. Dallas hears the sound of the stealth drive but does not identify it as a submarine. Putting information about Ramius' letter together with the subsequent launch of the entire Northern Fleet, Ryan deduces Ramius' plans. The U.S. military reluctantly agrees, while planning for contingencies in case the Soviet Fleet has intentions other than those stated. As tensions rise between the U.S. and Soviet fleets, the crew of Dallas analyze sonar tapes of Red October and finally realize that it is the sound of a new propulsion system. Ryan must contact Ramius to prevent the loss of the submarine and her decisive technology. After it is revealed that Ramius has informed Moscow of his plan for him and his officers to defect, Ryan becomes responsible for shepherding Ramius and his vessel away from the pursuing Soviet fleet, and meets with an old Royal Navy acquaintance, Admiral White, commanding a task force from the aircraft carrier HMS Invincible.
In order to convince the Soviets that Red October has been destroyed, the U.S. Navy rescues her crew after Ramius fakes a reactor meltdown. Ramius and his officers stay behind, claiming they are about to scuttle the submarine to prevent it getting into the hands of the Americans. A decommissioned U.S. ballistic missile submarine, the USS Ethan Allen, is blown up underwater as a deception. A depth gauge taken from the main instrument panel of Red October (with the appropriate serial number) is made to appear as if it was salvaged from the wreckage. Meanwhile, Ryan, Captain Mancuso and some of his crew, and Owen Williams (a Russian-speaking British officer from Invincible) board Red October and meet Ramius face-to-face.
The deception efforts succeed in convincing Soviet observers that Red October has been lost. However, GRU intelligence officer Igor Loginov, masquerading as Red October's cook, is aware of what Ramius is doing and attempts to ignite a missile rocket motor inside a launch tube so as to destroy Red October. Loginov opens fire with his weapon, killing Captain Lieutenant Kamarov (the ship's navigator) and seriously wounding Ramius and Williams. Ryan attempts to persuade the fiercely patriotic Loginov to surrender rather than die in the explosion, but Loginov refuses. Ryan manages to kill Loginov in the submarine's missile compartment.
Captain Viktor Tupolov, a former student of Ramius and commander of the Soviet Alfa-class attack submarine V. K. Konovalov, has been trailing what he initially believes is an Ohio-class vessel. Based on acoustical signature information, Tupolev realizes that it is Red October, and proceeds to pursue and engage it. The two U.S. submarines escorting Red October are unable to fire due to rules of engagement, and Red October is damaged by a torpedo from the Alfa. After a tense standoff, Red October rams Konovalov broadside and sinks it.
The Americans escort Red October safely into dry dock in Norfolk, Virginia, where Ramius and his crew are taken to a CIA safehouse to begin their Americanization. Ryan is commended by his superiors and flies back to his posting in London.

Question: Why does the Soviet Northern Fleet claim to be following the Red October?
Answer: A search and rescue mission.
Alternative answer: To search for and rescue the Red October.