Always means that the container will be restarted even if it exited with a zero exit code (i.e. successfully). This is useful when you don't care why the container exited, you just want to make sure that it is always running (e.g. a web server). This is the default.

OnFailure means that the container will only be restarted if it exited with a non-zero exit code (i.e. something went wrong). This is useful when you want accomplish a certain task with the pod, and ensure that it completes successfully - if it doesn't it will be restarted until it does.

Never means that the container will not be restarted regardless of why it exited.

These different restart policies basically map to the different controller types as you can see from kubectl run --help:

--restart="Always": The restart policy for this Pod. Legal values [Always, OnFailure, Never]. If set to 'Always' a deployment is created for this pod, if set to 'OnFailure', a job is created for this pod, if set to 'Never', a regular pod is created. For the latter two --replicas must be 1. Default 'Always'

And the pod user-guide:

ReplicationController is only appropriate for pods with RestartPolicy = Always. Job is only appropriate for pods with RestartPolicy equal to OnFailure or Never.