What does SSN mean for us navy submarines?

The hull designator denotes the hull type, it's purpose, and its propulsion mode. For example, SSN stands for "Submersible Ship, Nuclear"; SSBN is Submersible Ship, Ballistic, Nuclear; and SSGN is Submersible Ship, Guided Missile, Nuclear. All nuclear powered vessels, submarine or surface, are denoted with an "N" in the designation.

"SS" for "Submersible Ship", goes back to the earliest days of Submarines, when they weren't true submersibles like they are today. Until teardrop and cigar shaped hulls were designed, and longer range propulsion methods made crew limitations the only liability, older submarines were essentially surface vessels that had a limited submerged operational capability. They were, literally, "Submersible Ships". What makes them different than today's boats is that the old design made them much faster on the surface than the ever were submerged. Today's submarines are true submersibles by all definitions, much faster submerged than when surfaced.

For submarine hull types, there are currently 3 in the U.S. inventory - SSN, SSGN, and SSBN. SSN's are Nuclear Fast-Attack Submarines - L.A., Seawolf and Virginia Classes; SSBN's are Ballistic Missile Submarines (Ohio class, etc.); and SSGN's are converted older SSBN's that have had their Ballistic Missiles removed as part of the Treaty with the former Soviet Union. They were converted to Cruise Missile launch platforms.

During the Korean War, many former WW2 Fleet Boats were converted for Radar Picket duty along the Korean coastline. Their designations were "SSR", for "Submersible Ship, Radar Picket"