Project

A project manages one git repository and adds an issue tracker, a wiki, etc:

See: https://docs.gitlab.com/ee/user/project/index.html#doc-nav

Groups

Groups can be used to combine several projects under one topic. Examples:

You could put all your JavaScript projects in a group called "js"
You could create one group for each of your developer teams; then if somebody new joins a team, you can just add them to the group instead of having to grant developer access for every project.
Groups can be referred in issues or commit discussions.

See: https://docs.gitlab.com/ee/user/group/index.html#doc-nav

Different groups may have different permissions

Subgroups

Subgroups are only available in GitLab 9 and above; they allow you to create additional groups inside of groups, e.g. "js/frontend"; this is useful if you are working on a large project with several sub-projects. See the official documentation for some examples:

https://docs.gitlab.com/ee/user/group/subgroups/index.html