Dependencies

These are the packages that your package needs to run, so they will be installed when people run

 npm install PACKAGE-NAME

An example would be if you used jQuery in your project. If someone doesn't have jQuery installed, then it wouldn't work. To save as a dependency, use

 npm install --save

Dev-Dependencies

These are the dependencies that you use in development, but isn't needed when people are using it, so when people run npm install, it won't install them since the are not necessary. For example, if you use mocha to test, people don't need mocha to run, so npm install doesn't install it. To save as a dev dependency, use

npm install PACKAGE --save-dev

Peer Dependencies

These can be used if you want to create and publish your own library so that it can be used as a dependency. For example, if you want your package to be used as a dependency in another project, then these will also be installed when someone installs the project which has your project as a dependency. Most of the time you won't use peer dependencies.