iOS 10 Today Widget Buttons

Today Extensions have an NSExtensionContext, exposed as property extensionContext. ExtensionContext has an openURL method, which you can pass a private URL to launch your app.

This is a pretty broad question, so providing a broad overview of the technique.

(1) register a custom URL scheme in your main app, e.g. myapp:

Add keys to application plist file, e.g.:

<key>CFBundleURLTypes</key>
<array>
 <dict>
  <key>CFBundleURLName</key>
  <string>com.mycompany.myapp</string>
  <key>CFBundleURLScheme</key>
  <array>
   <string>myapp</string>
  </array>
 </dict>
</array>

this registers with iOS that your application handles the "myapp" scheme. Use your own name here, and pick one that is highly likely to be unique!

(2) call openURL on your extensionContext of your today extension view controller

[self extensionContext] openURL:[NSURL URLWithString:@"myapp://someurl"] completionHandler:nil];

(3) handle the call in the Application Delegate of your app and it's handleOpenURL method

-(BOOL)application:(UIApplication *)application handleOpenURL:(NSURL *)url
{
   // do something here
}

Of course this assumes you've placed buttons in your today widget and have the appropriate handlers to consume the button press and pass an appropriate URL to the app.

You've tagged the question with Swift, but these sample snippets are in Obj-C, sorry about that.