I'm planning on starting a social media website similar to other large social media websites. Do I need to register it as a business in order to make profits from it and/or get a trademark for its name? (I live in Georgia, USA)
In general, no. You can run a business as a "sole proprietor." However, certain activities may require you to get a business license.
For example, if you need to collect state or local sales tax because you are selling taxable things, you probably need to register with whoever collects the tax. If you are hiring employees, you generally need a federal Employer Identification Number (EIN) so that you can withhold income tax, pay FICA deductions, etc. If you are going to have a particular place that is open to the public for business, you may need to register with your state or local government. Depending on your line of work, you may need to get some sort of license or permit (health inspection, safety inspection, etc).
If all you are doing is running a website and getting a trademark, then there is no legal requirement that you incorporate, create an LLC, or register as a business. However, there are some good reasons to do so. For example, if you don't make a business entity (corp or LLC) separate from yourself, then if someone wants to sue your website, they would have to sue you personally, and you could lose money. If you operate the corp or LLC, then typically you can not personally be sued for your business activities.
Many developers start by running a website or an app without a corporation, then when any significant business activity or public presence begins, they create one. An accountant or lawyer can advise you further on your specific case.