Barostatting means controlling the pressure in an MD simulation. Thermostatting means controlling the temperature of the particles. Since the pressure includes a kinetic component due to particle velocities, both these operations require calculation of the temperature. Typically a target temperature (T) and/or pressure (P) is specified by the user, and the thermostat or barostat attempts to equilibrate the system to the requested T and/or P.
Barostatting in LAMMPS is performed by fixes. Two barostatting methods are currently available: Nose-Hoover (npt and nph) and Berendsen:
The fix npt commands include a Nose-Hoover thermostat and barostat. Fix nph is just a Nose/Hoover barostat; it does no thermostatting. Both fix nph and fix press/berendsen can be used in conjunction with any of the thermostatting fixes.
As with the thermostats, fix npt and fix nph only use translational motion of the particles in computing T and P and performing thermo/barostatting. Fix npt/sphere and fix npt/asphere thermo/barostat using not only translation velocities but also rotational velocities for spherical and aspherical particles.
All of the barostatting fixes use the compute pressure compute to calculate a current pressure. By default, this compute is created with a simple compute temp (see the last argument of the compute pressure command), which is used to calculated the kinetic component of the pressure. The barostatting fixes can also use temperature computes that remove bias for the purpose of computing the kinetic component which contributes to the current pressure. See the doc pages for the individual fixes and for the fix_modify command for instructions on how to assign a temperature or pressure compute to a barostatting fix.
As with the thermostats, the Nose/Hoover methods (fix npt and fix nph) perform time integration. Fix press/berendsen does NOT, so it should be used with one of the constant NVE fixes or with one of the NVT fixes.
Thermodynamic output, which can be setup via the thermo_style command, often includes pressure values. As explained on the page for the thermo_style command, the default pressure is setup by the thermo command itself. It is NOT the pressure associated with any barostatting fix you have defined or with any compute you have defined that calculates a pressure. The doc pages for the barostatting fixes explain the ID of the pressure compute they create. Thus if you want to view these pressures, you need to specify them explicitly via the thermo_style custom command. Or you can use the thermo_modify command to re-define what pressure compute is used for default thermodynamic output.