How you join a group depends on how the group is configured.
Groups that are configured as "joinless" or "private unlisted" do not support requesting membership at all. For "private listed" groups, the group membership request form is presented immediately when visiting the group. For other types of groups, a banner will appear towards the bottom of the web browser window that says the group "requires membership for participation - click to join." Clicking this message will either allow you to join immediately or present you with a form to make a formal request.
If you must request membership, you can specify a message to be sent with the request that will be shared with the group owners. You will then need to wait for the group owners to review your request and accept it. You will receive a notification when the request has been reviewed.