Hi, 1080p videos are still a bit problematic (but usually possible). There are 2 HD video acceleration APIs for Linux : VDPAU (from nvidia) and VA-API (desined by intel, but meant to be used anywhere : for instance VDPAU can be used through VA-API). A few days ago, Intel released a driver update
which should enable accelerated h264 (usually used to encode HD) playback.
Thats for the driver part.
media player wise, you'll need a media player that supports va-api in your case. mplayer supports it through an external patch, VLC 1.1 supports it out of the box.
In short, you might have to upgrade your graphic drivers (not 100% sure when va-api support came along) which would also mean upgrading your kernel ,librm and maybe some other things.
You will have to upgrade your media player. I suggest VLC 1.1
Another possibility would be trying the multithreaded version of mplayer (it helped a lot one one of my PCs)
Disclaimer : I don't use ubuntu, so I don't know what patches they apply where. to begin with, search for anything va-api (libva) related in the package manager, it might just work, though I doubt it.
Edit : forgot to mention flash
the official flash player doesn't support any hardware aceleration on linux as far as I know, the open source player Gnash does.