What is a VPS and when do you need one?
The short answer
A VPS is your own virtual server, with guaranteed capacity and full control, from a few dollars a month. You need one when shared hosting starts pinching, whether that means running custom software, handling serious traffic, or hosting several sites. Self-managed means you need some technical skill.
In detail
On a VPS you get a walled-off share of a physical server, with its own operating system, memory, and CPU. Nobody can eat your capacity and you can install whatever you want, whether a specific database version, custom software, or several sites side by side. DigitalOcean and Hetzner sell such servers from a few dollars a month, with Hetzner delivering remarkable power for the money.
The real price is responsibility. Updates, firewall, security, and backups are your job. If you want VPS power without system administration, the middle path exists. Cloudways puts a management layer on top of cloud servers, so you get the muscle without being a sysadmin.
When is the moment? When your site structurally hits shared hosting's limits, when you want to run something beyond a standard site, or when you are a developer who simply wants the freedom. Until then, good shared or managed hosting is the wiser choice.