@hansw circles is a permissions thing.. Before circles you could either post publicly, followers only, direct, or unlisted. Circles allow you to restrict the permission on viewing a post to a specific set of followers. Think of it like a direct message but instead of needing to list the people who can see it by tagging them in the post you can now limit it based on circles. So only people in those circles can access it or see it at all, but they wont get notified as they might with a mention. So no harm done if your circles are very large, its just followers only but restricted to a smaller scope.

Groups are entirely different. A group is a special account such that if you make a post and mention the group account in a tag the group account will repost your toot from the group account. this way anyone following the group account will see messages from anyone who happened to tag the group. This allows people to join groups by following them and posting to the group by tagging the group. Its similar in concept to groups on facebook where members of the group can add posts shared by the group.