I suppose there are a couple other ways to interpret your question, ikotabas. First, Flash is not used to design a website -- it is a kind of widget that can be used on a website to provide some sort of functionality, such as banners or games. Second, like everything else in computer science, there are things Flash is good for and things that it can be bent to do but which something else would probably be better suited to. Flash-based website elements are usually relatively large (file size, not necessarily screen size), so things like banner are probably best done using animated GIFs or static JPGs. If you are designing a new site that will include Flash-based games, then you can safely simply reserve some preset size (640x480 would probably work most of the time) for the flash games on whatever page you want the games served from.
IMHO: It IS possible to create an entire website using Flash
{examples}. However, unless you are going for artsy or some specific goal that the power of Flash is necessary for, DO NOT DO THIS! It is a lot harder than it looks to do this well and, depending on how efficiently it is done (likely not very if this is one of your first websites), it can require a substantially powerful client machine and broadband connection in order to work smoothly.