The Version Installer on the Tools page changes which Minecraft version and server type your server runs. It covers Vanilla, Paper, Fabric, Forge and the other supported types.
Your world is not deleted by a version change. The installer replaces the server files, not your save. That said, take a backup first — it takes a few seconds and it is the difference between an inconvenience and a loss.
Going up a version is usually safe. Going down is not. Once a world has been opened in a newer version, its files are rewritten in that version's format. Loading it on an older version can corrupt it or fail outright. If you need to go back, restore a backup taken before the upgrade.
Mods and plugins do not carry over. A plugin built for one version will usually refuse to load on another, and mods are stricter still — a Fabric or Forge pack is tied to the exact version it was built for. After changing version, expect to update them.
The install runs in the background and takes a few minutes. The server cannot be started until it finishes. Once it does, start the server and check the console for the version line before inviting players back.
If you are switching to a modpack rather than a plain version, use the modpack installer instead — it handles the two-step install that packs require.
If you need help with this, please submit a ticket.