To expand on Ashwin's answer, there are a few things you can do about this:
1. Check on the Adobe Flash website and download the latest version of Flash available for your (very out-of-date) system. You didn't indicate if your machine could be upgraded to a newer OS, but that's another thing you should probably put on your checklist to do. If you're incredibly lucky, you might have one of the few Intel Macs running Tiger, and you could upgrade to Snow Leopard perhaps.
2. If you're already at the max for Flash, try switching your preference to the HTML5 player instead. That should work for you.
YouTube ... most other video sites (like Vimeo) can also have a pref set for using HTML5, but forget (on all video sites) about watching videos in HD. Not on that machine.
3. Even if your version of Flash is working, it may be working poorly due to a nearly-full disk or a severe need for maintenance. Check to see if you have at least (dead minimum) 12GB or so free space on your HD, and download the 10.4 version of Maintenance (link conveniently provided
here, it's free) and just check every box and run it.
4. Finally, part of the problem you are seeing here (of which one of the above should mitigate the issue) is a function of age. A Mac that runs Tiger is probably at least six years old, maybe quite a bit older, and likely to be a PPC machine. What that means is that there isn't much in the way of upgrades or new programs for such machines, and as technology improves in various ways you're going to run into a similar problem more and more often. To put it another way, time to start thinking about getting a new(er) machine that can handle modern requirements before the experience gets to be really frustrating.
If you're not in the market for a brand new machine (though I'd encourage you to look at the Mac mini or iMac line), anything from 2009 and up should be handle everything today's technology throws at it, apart from some very top-end games or other super-demanding software. I've got a 2009 MBP I'm using, and I expect it will handle my varied needs for another year or two before it requires replacement.