There are many different ways to host your side project. The best choice will change depending on the project and how it needs to scale, but to get started any of the following may work...
Shared hosting
Shared hosting is a type of hosting service where multiple websites are stored on a single physical server.
Your website or application shares the same server resources as many other websites. As a result, it is typically the most affordable hosting option.
You'll have access to limited resources (CPU, RAM, etc.) and these are shared with all users.
The hosting provider handles server maintenance and security.
Suitable for beginners with user-friendly control panels. Users have restricted access to server configuration.
Website performance can be affected by other sites on the server.
Shared hosting is suitable for small websites, blogs, or businesses with low to moderate traffic. However, it may not work for large, resource-intensive websites.
Virtual Private Servers
A step up from shared hosting, VPS offering more control and resources.
A physical server is divided into multiple virtual servers, each acting as an independent unit.
Each VPS has is allocated its own CPU, RAM, and storage. Resources can often be easily upgraded as your needs grow. Performance of one website is less likely to affect others on the physical server.
Users have full control over their server environment, as a result you can install your preferred operating system and software. Each VPS is isolated from others, enhancing security and stability.
You are more responsible for the configuration of your VPS, as such you'll need more technical knowledge.
Dedicated Servers
Similar to a VPS, with dedicated servers you have an entire physical server dedicated solely to your website or application. You have access to all CPU, RAM, storage and bandwidth.
You can often customize the server to your needs, more RAM, less CPU or vice verse. These can typically be upgraded, with minimal downtime. Some providers support hot swapping of hardware, to eliminate downtime.
You have full root access and ability to customize the server environment, since no other users are on your server potential vulnerabilities are reduced.
As with VPS, you need server management skills to configure and maintain your server.
Since you have exclusive access to the hardware, you'll get great performance but at a significant price.
IaaS
A cloud computing model that provides Infrastructure as a Service.
IaaS offers virtualized hardware, such as virtual machines (VMs), storage, and networking capabilities, which can be scaled up or down based on the user's needs.
Where a VPS give access to individual servers, IaaS allows you to build an entire systems of private, connected resources for your application, each customized to your needs.
Amazon, Azure and Google and many other companies offer IaaS, systems they often for their own application hosting.
PaaS
Another cloud computing model, that rather than offering just infrastructure, provides a platform for developers to build, run, and manage applications. They handle the complexity of maintaining the underlying infrastructure.
PaaS provides a platform with tools and services designed for application development.
Allows you to focus on writing code rather than managing infrastructure.
Platforms can often automatically scale resources based on application demands.
This convenience can come at a significant price, PaaS is often the most expensive type of hosting.
Heroku was one of the first PaaS providers, but they have become very popular now. Cloud Run is available from Google. Digital Ocean offers its App Platform.