If you receive a message with the characters, either as an iMessage or an SMS text from contacts on other platforms, your iPhone will crash--but only if you open it. If you receive a notification with the message on your lock screen, your phone will either reboot or lock you out of Messages altogether.
The bug doesn't actually generate the message--some prankster with your phone number has to actually send the code to you. Let's hope your friends aren't jerks. If one of your contacts does send you a malicious message, its effects can be reversed with a follow-up message--or you can send yourself the fix from your Mac by replying to the original string in Messages on OS X, according to The Verge. (The bug only affects iPhones, iPads, and Apple Watches.)
You can also use Siri or a share sheet to send yourself a message on your iPhone or iPad, if you don't have a Mac or can't get to it.
The impact on you: This bug is incredibly annoying, but isn't the first time Messages has been hijacked. The only real solution: A software update from Apple. In the meantime, don't open messages with strange characters, and if your friends are jerks, turn off notifications for Messages in your iPhone settings (and do the same for Apple Watch) so the offending Unicode won't pop up on your lock screen. We'll update this story when the fix is in.