If you have a stop with no plane change, you will just stay on the plane or I think sometimes they'll let you roam the gate area for a few depending on how long the stop is.
If you are changing planes, it most likely means a gate change at a minimum. So you take your stuff, get off the plane, and go over to the new gate that your connecting flight leaves from. This could be the next gate over or even in a different terminal depending on the airport.
If you have a plane change, many times the flight attendants will announce the gates for each destination they know they have people connecting to. Keep your tickets handy, and you can always stop and ask the flight attendants either on your plane or at the gate.
It's not usually that terrible, though it's not fun to run through the airport.
Like the previous poster said, you are still security cleared, so you will be on, and remain on the airplane side of security.