That is a conversation that has become "pinned" Control+Click or right click on a conversation and a context menu will appear with the option to pin (or UN-pin) the conversation. If the person in the conversation is in your contacts list and has a photo associated with the entry that is the picture you will see. Otherwise it is a conversation icon. If you have a lot of conversations going it can be very useful to pin the most critical to the top and then you can drag the pinned conversations into whatever order mess the most sense to you.

If we knew what it was we were doing, it wouldn't be called research, would it?

— Albert Einstein