24p describes the refresh-rate of 24 progressive frames per second in the digital data.

Cinema film is exposed with 24 full frames per second, so 24p material is the exact digital equivalent of the original analog film material. Thus it's regarded as the native digital cinema-format.

Neither film in a cinema nor digital 24p material is projected at 24 frames per second.  A projection with such a low refresh-rate would result in heavy flickering because the eye needs a projection rate (=refresh-rate) of at least 40...50 frames per second to perceive a fluent, flicker-free picture. In a cinema this is achieved by projecting each frame twice and thus having a refresh-rate of 48 frames per second (or to be more precise 24x2 frames per second). For the same reason an electronic projector or display also has to display each frame of a 24p stream twice (or any other integer multiple) to produce a refresh-rate of 48 fps (or 72,96,...fps) to avoid flickering.

Basically it makes no difference if the frame-doubling from 24p to 48p is done by the HTPC through the video-codec or the graphics-driver or by the display-electronics. So the HTPC could send a 48p signal to the display if it is ensured that the frames are really doubled instead of inserting a motion-interpolated frame.  In the former case the cinema look-and-feel is preserved with the typical judder in fast horizontal motions or camera-pans.  In the latter case, horizontal motions and camera-pans are more fluent but lack the typical cinema-look.

There is also a third way to send 24p material to a display: Each frame is split into two interlaced half-frames that are sent with 48 fps.   Contrary to interlaced source material (e.g. from TV-cameras) here we have no motion difference between the two half-frames because they originate from the same full-frame. The display simply has to add each pair of half-frames to recreate two identical full-frames which are displayed with 48 fps. To preserve the correct "cinematic" display there may be absolutely no motion interpolating between the half-frames neither by the HTPC nor by the display-electronics. Some graphics drivers offer this special mode as "24i" (sometimes also shown as "24Hz mit Zeilensprung") which should be more precisely called 2x24i- or 48i-mode.

Further reading:

• Wikipedia article on 24p