Update npm 5:
As of npm 5.0.0, installed modules are added as a dependency by default, so the --save option is no longer needed. The other save options still exist and are listed in the documentation for npm install.
Original answer:
Before version 5, NPM simply installed a package under node_modules by default. When you were trying to install dependencies for your app/module, you would need to first install them, and then add them (along with the appropriate version number) to the dependencies section of your package.json.
The --save option instructed NPM to include the package inside of the dependencies section of your package.json automatically, thus saving you an additional step.
In addition, there are the complementary options --save-dev and --save-optional which save the package under devDependencies and optionalDependencies, respectively. This is useful when installing development-only packages, like grunt or your testing library.