The Git manual has as an exceptionally helpful description of stashing. They call it "Temporarily setting aside work in progress".
While you are in the middle of working on something complicated, you find an unrelated but obvious and trivial bug. You would like to fix it before continuing. You can use git-stash(1) to save the current state of your work, and after fixing the bug (or, optionally after doing so on a different branch and then coming back), unstash the work-in-progress changes.
$ git stash save "work in progress for foo feature"
This command will save your changes away to the stash, and reset your working tree and the index to match the tip of your current branch. Then you can make your fix as usual.
... edit and test ...
$ git commit -a -m "blorpl: typofix"
After that, you can go back to what you were working on with git stash pop:
$ git stash pop
Git's stashing feature saved me quite some trouble a couple of days ago, while working on a side project. I developed a feature on branch "B" and had quite a big change-set, while someone found a bug on the stable branch, let's call it "A".
Coming from SVN, I thought I had to check-in my changes in order to switch to Branch "A" and fix the bug. That's quite ugly, because it screws with your history of commits and even might break your builds. I know that there are other possibilities with SVN, like copying your changed files or creating a patch file. However, to me those all look like workarounds rather than solutions.
So I simply stashed my changes from branch "B" and got a clean branch with HEAD. I switched to branch "A" and was able to fix and commit the bug. After switching to branch "B" again, I simply poped the changes from the stash, as if I had never stopped working on the feature.
After working with Git for some time, I would really recommend people to switch to a DSCM. I'm not alone with this. SpringSource's DM server team migrated their repositories to Git a couple of weeks ago and they didn't hear any complaints from the community. No matter if you choose Git or Mercurial, the advantages of a DSCM really outweigh the initial costs of getting started.