Puppet 3.8.2 is a bug fix release (with future parser changes) in the Puppet 3.8 series.
The main focus of this release is to make sure the 3.8 future parser is forward-compatible with the Puppet language as of Puppet 4.2. It also add some new reserved keywords (if using the future parser) and it fixes several bugs.
Check out the release notes for more information:
https://docs.puppetlabs.com/puppet/3.8/reference/release_notes.html#puppet-382
You can see the full list of changes on the release's JIRA page:
https://tickets.puppetlabs.com/jira/secure/ReleaseNote.jspa?projectId=10102&version=13415
If you're installing Puppet for the first time, follow the Installation Guide: https://docs.puppetlabs.com/guides/install_puppet/pre_install.html