I can tell you what the Connection item is for. It's designed so that if you have information on the same contact in multiple places (for example, in your contacts, in twitter, in facebook and G+, you can link , or "connect" them together and see all the information consolidated into one virtual contact on your phone.
If you open a contact, and tap on the row with the chain link, it will open a page with that contact, and a button at the bottom to "Join another contact." Tap that, and you can search all your contacts, select the one you want to link, and then they'll be joined. If you tap the chain link on a contact that is already joined, you'll see all of the individual contacts, and be able to separate them, if you wish.
As for the problem, under the word "connection" should be a symbol that indicates where the contact is coming from (e.g., a "g" for Google, a square shape for Exchange Active Sync, a phone handset if it's locally stored on your phone, etc.) What symbol do you see for your contacts?