The idea behind a connected copy is to indicate that the item in each language is going to represent the same thing. For example, if you have a News Article in English and you were going to translate it to French, you should probably create it as a Connected Copy. This tells the content manager that both items represent the same thing (a single news article), but in different languages.
You'll notice two things about connected copies. First, if you're editing a content item and change the language using the dropdown at the top of the dialog, the connected item in the other language will be opened immediately, since that's the other-language equivalent of the item you already had open. Secondly, from a more technical standpoint, the item in both languages will have the same ContentID. So, if you have a ContentID (for example, from a query string) you'll know how to get that content in all languages in Agility.
Editing an item which has connected copies does not affect the content of the copies. Updating one won't update the others unless you explicitly make changes to the other items in the content manager as well. The exception to this is fields which have been designated as "Constant across all languages" in the content definition (default to false). This is useful for fields like Product Codes on Products which should be the same no matter what language the website is in.