Hosting (Putting Things on the Internet)
Putting a website on the internet is called web hosting. The server that stores the website is the web host. To serve a website or offer a service to any person on the earth you need a computer that is connected to the internet. This computer must run a web server application that is waiting to receive requests from clients, for instance web browsers. Also, it should be running all the time, because you don't know when someone wants to access your website. It could be in the day, night or from a different time zone.
Another thing to consider is the amount of visitors you expect (this is called traffic). While you might only expect moderate traffic when you start to come online, it can happen that a television broadcast, viral tweet, or forum post suddenly creates a high interest in your website. In that case, the flood of people might bring the server to its knees, freezing and breaking down on the high demand.
I would recommend you to rent a server in data center via a hosting platform (also called web hosting service provider or cloud provider) because configuring a computer at home as a server has multiple downsides. You should consider these downsides before considering the trade-off between the extra work and the cost of running a server at home compared to simply renting a server.
If you want to run a computer as a server you need to embed it in your local network (LAN or WLAN) and configure the network to be open for anyone on the internet. You also have to manage the security of the system and possibly some maintenance tasks like updating the operating system. Opening up your network to strangers like this, might accidentally create a security vulnerability which could expose the private data of another device in the network. A computer makes noise and blows out heat, which might be undesirable in a space you live in. Also, you need to buy the computer itself and make sure it is connected to an electrical outlet at all times (unplugging when vacuum cleaning should be prevented 😉). So, you pay for the hardware and ongoing electricity bill. It might be alright for a hobby project if you know for sure what you are doing and the costs are lower than renting. Other than that, I think it is not worth it because it is more difficult to scale things. What's scaling?
Horizontal vs. Vertical Scaling
When you rent a server the cheapest offerings are called virtual private server (VPS). This means one computer is running multiple operating systems in virtual machines. So, the resources of one computer are shared. You as a customer get one of these machines assigned. When one of the other virtual machine receives a lot of traffic and under heavy load this could slow down your own virtual machine. If you want better performance you can rent a dedicated server. In that case, there is no virtualization layer. You alone can use all the resources of the computer.
I don't have a lot of experience with the information below, and I don't have a lot of confidence in the information. So take it with a grain of salt. The general idea should be okay, though.
Now, what about scaling? Scaling means that you have higher performance requirements (e.g. due to increasing popularity of our web site and its traffic). In case of a VPS scaling is easy because you can reconfigure your virtual machine and assign more resources. For a dedicated server it gets a bit more complicated. There are two options: Either you can exchange the components of the computer to gain more more power (CPU, RAM, storage, etc.). This is called vertical scaling. Alternatively, you can add more computers and put a load balancing server in front of them to receive and distribute traffic across the different computers. This is called horizontal scaling.