TrueIP is a small free tool which can discover your external IP address, display it in various ways and alert you when it changes.
On launch the program adds an icon to your system tray. Hovering your mouse cursor over this displays a tooltip with your current external IP.
You can also display the IP by double-clicking the icon, or right-clicking it and selecting "Quick Information".
Right-clicking the icon and selecting "Full Information" displays your external IP, local IP, MAC address, host name and more.
An Options dialog enables launching the program at startup, using a different source to retrieve your address (the default is myip.dnsomatic.com), how often to check the address (every 10 minutes by default), and more.
If it's important to know when your address changes, you can have it displayed in a tray notification, sent in an email, saved to an FTP server, or just logged to a file for analysis later.