Why won't my mac play 1080p videos on youtube?

I suggest that you use Activity Monitor and monitor the download of your Network to determind if you are getting any data downloading, during your attempt to view a 1080p video.

My internet download speed is so slow that when I attempt to view a 1080p video, I have to wait a long time for the video to start, and then it will pause a long time, till enough data arrives to continue the display.

IMO youtube has gone down the, 'er, tubes. I think the problems with 720p, 1080p started happening around when google took it over and merged accounts. I use a fast Mac Pro, 5870 video card, lots of memory, 12meg internet connection, and even my machine has problems (frequent stalls with pinwheel spinning) with many (not all) youtube's 720p's and 1080p's. But I have no problems on any other sites (try Vimeo's HD for example).

So, bottom line, it is probably not your system. But you should check your Activity Monitor for any process hogs.