Git Stash vs Shelve in IntelliJ IDEA

git shelve doesn't exist in Git.

Only git stash:

• when you want to record the current state of the working directory and the index, but want to go back to a clean working directory.
• which saves your local modifications away and reverts the working directory to match the HEAD commit.

You had a 2008 old project git shelve to isolate modifications in a branch, but that wouldn't be very useful nowadays.

As documented in Intellij IDEA shelve dialog, the feature "shelving and unshelving" is not linked to a VCS (Version Control System tool) but to the IDE itself, to temporarily storing pending changes you have not committed yet in changelist.

Note that since Git 2.13 (Q2 2017), you now can stash individual files too.

In addition to previous answers there is one important for me note:

shelve is JetBrains products feature (such as WebStorm, PhpStorm, PyCharm, etc.). It puts shelved files into .idea/shelf directory.

stash is one of git options. It puts stashed files under the .git directory.

When using JetBrains IDE's with Git, "stashing and unstashing actions are supported in addition to shelving and unshelving. These features have much in common; the major difference is in the way patches are generated and applied. Shelve can operate with either individual files or bunch of files, while Stash can only operate with a whole bunch of changed files at once. Here are some more details on the differences between them."