Naming your site after your domain may seem obvious to some of you, but you'll be surprised to learn that not every website is named after the domain name even when the webmaster owns that domain name.
Naming a site after its domain name is important, for the simple reason that when people think of your website, they'll think of it by name. If your name is also your URL, they'll automatically know where to go. For example, when people think of anim.tv, they don't have to wonder what URL to type into their browser to get there. The name of the site is also the URL.
Imagine if your business (or website) is called "Acme", but somebody else holds that domain name. Instead, you have some obscure domain name called, say, "mybusiness.com". What happens when your customers, recalling that Acme has a product they want, type "www.acme.com"? They'll wind up at your competitor's website. One lost sale.
In the modern world of the Internet, where people automatically turn to the Web for information, it pays to have a domain name that reflects your site or business. There are just fewer things for your customers or visitors to remember. Moreover, you don't seriously think that they'll try to memorise an unrelated URL just because you want them to, do you? The only people who'll memorise it are you and your competitors who want to compare your prices.
What if you cannot get the domain name of your choice? It really depends on how committed you are to that particular name. If you have an existing brand name that you're known for, you'll probably not want to ditch that name just because you couldn't get the domain name. After all, it took you a lot of time and money to establish that name. If so, you might simply want to try to buy over the domain name from the current owner. Check up the "whois" information for the domain, and contact that person listed to see if they're willing to sell it. You probably should be aware that they are likely to want to charge a higher fee than you'll normally get when buying new domains (assuming they want to sell it in the first place).
On the other hand, if you're just starting out, you might prefer the cheaper alternative of trying to obtain a domain name first, and then naming your website (or business) after the domain that you've acquired. So if you've acquired, say, the domain name "acme.com", then your website and business might be named "Acme" or "acme.com". I know this seems a bit like putting the cart before the horse, but that's the reality if you don't want to lose out on the Internet.