Absolutely; not only are many of the ships and submarines real, the tactics were as well. I've been to many of the places that Clancy mentioned in his novel and that were referenced in the film. The Sonar Room aboard the DALLAS was total fiction though; during Battlestations, there are nearly a dozen Sonarmen crammed in a space about 12'x8', and the waterfall Sonar displays in the Red October's control room were more accurate for older Sonar systems of that era.
The USS DALLAS (SSN-700) was the actual fast-attack used in the movie; my old friend and Sea Puppy from USS RAY (SSN-653), and current author Donnie Meadows (http://www.authorsden.com/visit/author.asp?id=40920) was one of the lead Torpedoman on the DALLAS when they were filming "Red October" aboard ship. He had some interesting stories to tell.
The language used onboard was absolutely accurate, and that was due to the DALLAS crew advising Scott Glenn on the proper terminology to use. "Down Ladder" and "Make a Hole" are 2 terms you'll hear daily on any U.S. submarine.
As has been stated, the Red October was just sets, though not entirely inaccurate given what we knew about the Typhoon-class boats back then. All of the underwater sequences were CGI, but based on realistic submarine tactics used today, save for one or two.
Since the success of "Top Gun" in the 80's (which used real planes from Top Gun, then at NAS Miramar (now MCAS Miramar since they've moved it)), the Navy and other services realized the potential recruiting value in officially participating in military themed movies. That's why you've seen a lot of participation from the military advising on many big movies with military action sequences featuring real hardware. It's a cheap recruiting tool, and Hollywood gets the use of realistic hardware and accurate advice on tactics.
However, make no mistake; they will officially distance themselves from any movie that's BS or shows the military in a bad light, or is pure fiction. That's why the movie "Crimson Tide" was never sanctioned by the U.S. Navy, and also why it is one of the most inaccurate and BS submarine movies ever made about modern submarine warfare.
The USS MYSTIC (DSRV-1) was the DSRV referenced in the book and film, though the submerged sequences were CGI.
I still remember reading the novel on my first operational deployment to the North Atlantic in '81, wondering how the hell Clancy got all of his information, and why they let him publish it (it was published by the Naval Institute Press, so it cleared security review). He got a lot of it wrong, but he also got a hell of a lot right.
A couple of errors that were noted during the movie that doesn't jive with real life:
1. The torpedo dropped from the helo was an exercise Mk 48 torpedo; you can always tell and exercise shot as it's warhead is colored fluorescent orange to make it easier to find.
2. Though Russian acoustic homing torpedoes aren't as accurate or stubborn as ours, they're not as easy to evade either, just like ours. While the DALLAS' maneuvering in front of the torpedo to get it to switch targets was colorful, it's total BS; most modern torpedoes exceed 50 knots when in full acquisition and homing mode, and your a$$ is toast if it locks on, countermeasures notwithstanding. A couple of oversize Alka-Seltzer tablets aren't going to mask the noise of an L.A.-class boat at flank speed.
Probably the best recruiting movie of late is "Act of Valor". If you watched it, during the extraction scene in the water where the SWCC's are blanketing the area with minigun fire, they're actually using live rounds. Except for a few actors, all of the military participants were active duty SEAL's, SWCC's, or other Navy personnel.