1. dependencies
Dependencies that your project needs to run, like a library that provides functions that you call from your code.
They are installed transitively (if A depends on B depends on C, npm install on A will install B and C).
Example: lodash: your project calls some lodash functions.
2. devDependencies
Dependencies you only need during development or releasing, like compilers that take your code and compile it into javascript, test frameworks or documentation generators.
They are not installed transitively (if A depends on B dev-depends on C, npm install on A will install B only).
Example: grunt: your project uses grunt to build itself.
3. peerDependencies
Dependencies that your project hooks into, or modifies, in the parent project, usually a plugin for some other library or tool. It is just intended to be a check, making sure that the parent project (project that will depend on your project) has a dependency on the project you hook into. So if you make a plugin C that adds functionality to library B, then someone making a project A will need to have a dependency on B if they have a dependency on C.
They are not installed (unless npm < 3), they are only checked for.
Example: grunt: your project adds functionality to grunt and can only be used on projects that use grunt.