My understanding is that, once calibrated and profiled (using commercial hardware and software), a program automatically engages the monitor profile at boot time (usually by means of a profile loader installed as part of the profiling operation). Thus the monitor profile is active for whatever software is currently used to display an image.
Calibration and profiling are two different processes that shouldn't be confused.
Calibration standardizes the monitor response to a certain set of basic parameters - temperature, gamma and luminance, neutral color balance. It's just a linear 1D correction, very basic. It affects everything, system-wide, because it modifies the monitor itself.
Profiling builds a monitor profile. This is a full and complete description of the monitor's behavior,
in its now calibrated state, in three-dimensional color space. The profile description includes the position of the three primaries in color space, so it accounts for the monitor's gamut (wide or standard), as well as any other peculiarities and quirks in the monitor's behavior.
In short, the profile has a much higher precision level. But it is not system-wide, it is used only by some applications.
Color managed applications like Photoshop or Firefox convert the document profile to the monitor profile and sends it off to the display. A non-color-managed application, like Internet Explorer, just sends the numbers straight through without any correction for the monitor.
This works the same way as any color management operation: source profile > destination profile. That conversion is the color management part. The calibration LUT that is loaded into the video card/monitor is not a part of the color management chain.