About
A repository potentially holds multiple variants (tag) of an image.
Official repository are a carefully curated set of Docker repositories supported by Docker, Inc.
Automated repositories are Automated Builds that allow you to validate the source and content of an image. Automated Builds automate the building and updating of images from GitHub or Bitbucket, directly on Docker Hub. It works by adding a commit hook to your selected GitHub or Bitbucket repository, triggering a build and update when you push a commit.
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Example
In the case of the ubuntu image, there is multiple variants covering Ubuntu 10.04, 12.04, 12.10, 13.04, 13.10 and 14.04. Each variant is identified by a tag and you can refer to a tagged image like so:
ubuntu:14.04
List
docker images
REPOSITORY TAG IMAGE ID CREATED SIZE
docker-whale latest 1351cae1fdfb About a minute ago 275.1 MB
ubuntu latest f753707788c5 7 weeks ago 127.2 MB
hello-world latest c54a2cc56cbb 5 months ago 1.848 kB
docker/whalesay latest 6b362a9f73eb 18 months ago 247 MB
training/webapp latest 6fae60ef3446 19 months ago 348.8 MB