FSS stands for Flight Service Station. The FSS' primary role is to provide weather briefing and flight planning services to pilots. When you call 800-WX-BRIEF, you talk to a briefer at a Flight Service Station.
Airborne, you can call on the radio to FSS to get updated weather, file a PIREP, open/close VFR flight plans, and in some locations get direction finding assistance if you are lost (This is called a DF Steer).
If you're on the ground at an uncontrolled airport and unable to reach ATC on the ground when departing IFR, you can also call FSS to get your IFR Clearance relayed. The FSS won't give you a clearance, however they can pass one on from the ATC facility that is controlling.