One of of the long-time issues with using flash or designing flash websites is that search engines have a hard time reading flash. There have also ben a TON of annoying websites made in flash. The web designers that make these sites think that they are really cool and innovative, but when it comes down to it, most of the "flashy" designs are just annoying and don't do anything for the user except annoy them.
Flash is also very limited on mobile phone web design. Most lower-end phones have text-only browsers and most web designers who have used flash over the past 10 years have not taken the time to make their flash accessible. The ipad and iphone have also pretty much killed flash because their lack of support for it.
Flash has proven useful for other purposes though, such as video. The popularity of YouTube has made flash an essential aspect of video encoding. Web video games also commonly rely on flash for their funcitonality.
So where does HTML5 come into play? Well, it's basically providing funcitonality that does pretty much everything that flash could do. For more information about HTML 5 visit HTML5rocks.com