The simulator is just an application, and as such you can run it like any other application.
To open it in Spotlight, search for Simulator.
To run the simulator straight from terminal, prepend these locations with the open command.
Xcode 7.x, 8.x, 9.x, 10.x, 11.x, 12.x, and 13.x
In Xcode 7.x – 13.x, the iPhone Simulator has moved again: /Applications/Xcode.app/Contents/Developer/Applications/Simulator.app.
Xcode 6.x
In Xcode 6.x, the iPhone Simulator has moved yet again, and now resides here: /Applications/Xcode.app/Contents/Developer/Applications/iOS Simulator.app.
Xcode 4.x, 5.x
In Xcode 4.x (through 4.5 on Mountain Lion) and Xcode 5.0.x on Mavericks, it lives here: /Applications/Xcode.app/Contents/Developer/Platforms/iPhoneSimulator.platform/Developer/Applications/
In my version of Xcode (4.5.2), I find it quite convenient to use the Open Developer Tool menu from either the dock icon or the Xcode menu:
Xcode 3.x
In Xcode 3.x, it lives here:
/Developer/Platforms/iPhoneSimulator.platform/Developer/Applications/iPhone Simulator.app
In some future version of Xcode, it will probably move again, it’s a squirrelly little app.