I'm sure many people will be able to expand and correct greatly... but here goes
it's a system that incorporates precise measuring equipment to measure the amount of movement of the aircraft (acceleration in a particular direction); it combines the data with a map to give information the pilot can use; it gives a very accurate picture of where you are; the entire system is self contained, it dosen't need satellites or ground based signals to operate, however over time it accumulates error (much like the HI), and needs to be calibrated over a specific point (like a VOR).
INS is a closed loop nav system. It is guided by gyros, on upscale units, these are laser ring gyros. You spin up the INS, allow it to stabilize (gyros spinning and stabilized) and give it your starting point. That's why you see at some hardstands a lat/long of that particular hard stand. On the ship, we plugged in a cable that was hooked up to the ship's INS (SINS) that told our INS where it was. It then uses the percieved motion to navigate. Older units strictly work with Lat/Long, newer ones input that into the FMC/FMC and are synced with the nav database.
When we had alert launches, we often had the engines turning and everything ready to go (<5 minutes) but had to wait a few for the INS to stabilize. If it dumped in flight, you had to initialize with a GPS fix, a TACAN fix (VOR + DME from known point) or a RADAR fix (radar fix on known landmark, ie point of an island).
It's slick, it's expensive, and it's fairly transparent once you set it up.
Oh yeah...sometimes Doppler nav units are tied to INS's. Doppler nav shoots the equivalent of a RADAR wave at whatever you are flying over (ground, water) and gives the INS a secondary source of direction and speed to back up the gyros.
INS can be *very* accurate on it's own....INS's on subs operate for weeks without outside reference (GPS fixes) and are accurate well within a boatlength.