It depends on how the website makes money, but it’s usually a Gewerbe.
Commercial websites
Websites that are part of your business
If you are a Freiberufler, and your website is part of your business, it’s not a Gewerbe. You don’t need to register your website as a separate business. For example, a freelance writer can use their website to promote their services.
Websites that are a stand-alone business
If your website is a stand-alone business, and you make money by selling ads, referrals or products, it’s a Gewerbe.1 You must get a trade licence (Gewerbeschein) and register your business with the Finanzamt.
For example, if you make money by placing ads on your blog, you need to register it as a Gewerbe. An online store is also a Gewerbe.
Non-commercial websites
If your website is not used for any commercial activity, then you don’t need to declare it as a business. For example, you don’t need to register your personal blog or hobby project as a business if you don’t make money with it.