With the release of Watch OS 2, third-party developers are releasing watch app updates that include complication integration. Complications are the small bits of information, such as the current temperature or your activity progress, presented on a watch face without the need for you to open another app.
Prior to the release of Watch OS 2, complications were reserved for Apple's own apps. Think stocks, weather and calendar. But as developers continue to release updated apps designed for Watch OS 2, Apple's complications will quickly become outnumbered.
For example, Dark Sky, the weather app that alerts you when rain or snow is imminent, recently updated with a complication; as did American Airlines and Citymapper.
When an app update includes a watch complication, it's added to the list of available complications automatically. You'll still need to add it to your watch face, however.
You can check which apps offer a complication by launching Watch app on your iPhone, then tap on the new Complications category. A list of apps that offer a complication are found here. This is also where you'll need to go in order to edit or remove complications from the list of options.
Either drag an app down to the bottom "Do Not Include" section, or tap on the red circle next to the app's name. I've yet to figure out what reordering complications does in this section, as they're in alphabetical order on your watch.
Force Touch on your watch face, then tap on the Customize button to edit the complications displayed on your watch. Using the digital crown, scroll through the available complications; press the digital crown when you're done.