Don't get too bogged down in DirectX, chances are you can already play your video - but it's not clear if you've even tried or not yet?
Have you worked out how to get the video off the SDHC card and onto your computer (ie. using a card reader) or do you need help with that?
Do you have software that can play the video? (does the Panasonic software have a player).
As you asked, DirectX is a Microsoft component of Windows - if you have trouble sleeping

have a read of:
DirectX - Wikipedia, the free encyclopedia
Vista on the laptop will have DirectX 10 included, so it's likely that it's graphics card is compatible (if the laptop came with Vista installed, it's an almost-certainty).
Things may be less clear-cut with the desktop and it mind depend on it's age (and age of the graphics card). Chances are it's running DirectX 9 (and if you have Windows automatic updates turned on, it almost certainly will have). With regards to the graphics card, to find out which version of DirectX it's compatible with, best to either a) look in the manual/specs sheet or b) look on the manufacturer's website.
By the way, if you can't get your video to play, it's much more likely to be either a) a codec issue or b) graphics card firmware on the desktop needs updating, rather than a DirectX issue.