Unless you are starting up with a mega porn website you probably could get almost any webhost, even free. But once it takes off, a professional web-hosting provider will be required to cope with the high traffic volume and bandwidth usage. There are also hosts that only specialize in adult sites.
In addition, depending on the scope of your content, you may want to host abroad, with an offshore web-host. This can limit your potential legal liability and reduce the risk of getting your website taken down.