This page will demonstrate the easiest way to upgrade your InspIRCd servers with near minimal downtime. Please note, this is the easiest way. There are potentially more ways to upgrade your server, but are rather unnecessary to go into here.
Patch level upgrades
A patch level upgrade is when the third number in the version changes. For example, upgrading from 2.0.1 to 2.0.3. Upgrades such as this should not involve configuration file changes, and may be as simple as reloading a module.
First, identify the upgrades you will need or may want. If it is a module you want upgraded, simply upload the newest module version to your server and rebuild InspIRCd like normal. When the build completes, reload the module. Updates complete.
If, however, you need a bugfix in the core, or if you want to completely upgrade to a new patch level, your server will require a restart. To do this, download the copy of InspIRCd you are planning to upgrade to. Next, copy your InspIRCd configuration directory (usually ~/inspircd/conf) to a safe place outside of the InspIRCd directory. Remove your InspIRCd directory (you may want to shut down your server first). Now, simply extract the InspIRCd tarball of the newest version and build InspIRCd like normal. When InspIRCd finishes building, move your old configuration directory back to where it belongs and viola; you've upgraded your server.
Version level upgrades
A version level upgrade is when one of the first two numbers in the version changes. For example, 2.0 to 2.1 is a version level upgrade. Upgrades such as this will involve new features, deprecated or removed modules, and changes to the configuration file.
Version level upgrades are best done from scratch; the old configuration files probably won't be useful in the new version. You should, however, keep them around for reference and copypasta. Go through the new example configuration for new options or changes from the old version, and reconfigure InspIRCd's config files from scratch. If you're feeling reckless, try copying your old configuration file and using it for the new version. If it works, more power to you, but don't expect a happy support channel if it doesn't.