The lines, and when they actually run
Eight lines stop at Fulton Street itself. The MTA counts nine through the Fulton Center complex once you include the R at Cortlandt Street, which connects underground. Service isn't identical around the clock: the 2, 4, A and J stop here at all times; the 3, 5 and C stop at all times except late nights; the Z runs rush hours only, in the peak direction.
Which one you want
4 and 5: up the East Side. Brooklyn Bridge City Hall, then Union Square, Grand Central, 86th Street. The fastest way north if you're going to the East Side.
2 and 3: up the West Side, through Chambers Street and Times Square to the Upper West Side. This is the one for Times Square without a transfer.
A and C: Eighth Avenue northbound, Brooklyn southbound. The A is the express and the one that connects to JFK at Howard Beach.
J and Z: east over the Williamsburg Bridge, through the Lower East Side and out into Brooklyn and Queens.
The station is big and the transfers are long corridors rather than a single platform, so give yourself an extra couple of minutes if you're changing lines with luggage.
How close it is to us
About three minutes on foot. We're at 17 John Street, and Fulton Street station sits a couple of blocks north of us. That's the practical reason we tell people they don't need a car here: from one station you can reach most of Manhattan without transferring, and if you'd rather be on the water, Pier 11 at Wall Street is a short walk south.
A note on the Fulton Center itself
It opened in 2014 and it's less of a subway station than a mall with trains under it: a glass dome, shops, and passages that run west to the Oculus and the PATH. When it's raining you can cover a good chunk of the walk between the World Trade Center and Fulton Street underground.