When Siri came around with the iPhone 4S, we were then able to give ourselves a nickname for her to call us. This nickname then got added to our contact card. If you shared that contact card with anyone, they'd then have your nickname attached to it. Ironically, if you have nicknames enabled in iOS 7, that's the name that will show up for them in message threads, their Contacts, and any other communication you have with that person. In some cases, nicknames can be quite silly and you may not want that displaying for a certain contact.
Luckily you can completely disable iOS from using nicknames. Here's how: