How to handle with a default URL scheme

As you are mentioning AppleScript, I suppose you are working on Mac OS X.

A simple way to register and use a custom URL scheme is to define the scheme in your .plist:

<key>CFBundleURLTypes</key>
<array>
    <dict>
        <key>CFBundleURLName</key>
        <string>URLHandlerTestApp</string>
        <key>CFBundleURLSchemes</key>
        <array>
            <string>urlHandlerTestApp</string>
        </array>
    </dict>
</array>

To register the scheme, put this in your AppDelegate's initialization:

[[NSAppleEventManager sharedAppleEventManager]
    setEventHandler:self
        andSelector:@selector(handleURLEvent:withReplyEvent:)
      forEventClass:kInternetEventClass
         andEventID:kAEGetURL];

Whenever your application gets activated via URL scheme, the defined selector gets called.

A stub for the event-handling method, that shows how to get the URL string:

- (void)handleURLEvent:(NSAppleEventDescriptor*)event
        withReplyEvent:(NSAppleEventDescriptor*)replyEvent
{
    NSString* url = [[event paramDescriptorForKeyword:keyDirectObject]
                        stringValue];
    NSLog(@"%@", url);
}

Apple's documentation: Installing a Get URL Handler

Update I just noticed a problem for sandboxed apps that install the event handler in applicationDidFinishLaunching:. With enabled sandboxing, the handler method doesn't get called when the app is launched by clicking a URL that uses the custom scheme. By installing the handler a bit earlier, in applicationWillFinishLaunching:, the method gets called as expected:

- (void)applicationWillFinishLaunching:(NSNotification *)aNotification
{
    [[NSAppleEventManager sharedAppleEventManager]
        setEventHandler:self
            andSelector:@selector(handleURLEvent:withReplyEvent:)
          forEventClass:kInternetEventClass
             andEventID:kAEGetURL];
}

- (void)handleURLEvent:(NSAppleEventDescriptor*)event
        withReplyEvent:(NSAppleEventDescriptor*)replyEvent
{
    NSString* url = [[event paramDescriptorForKeyword:keyDirectObject]
                        stringValue];
    NSLog(@"%@", url);
}

On the iPhone, the easiest way to handle URL-scheme activation is, to implement UIApplicationDelegate's application:handleOpenURL: - Documentation

All credits should go to weichsel and kch

I'm just adding swift(2.2/3.0) code for your convenience

func applicationWillFinishLaunching(_ notification: Notification) {
    NSAppleEventManager.shared().setEventHandler(self, andSelector: #selector(self.handleGetURL(event:reply:)), forEventClass: UInt32(kInternetEventClass), andEventID: UInt32(kAEGetURL) )
}

@objc func handleGetURL(event: NSAppleEventDescriptor, reply:NSAppleEventDescriptor) {
    if let urlString = event.paramDescriptor(forKeyword: keyDirectObject)?.stringValue {
        print("got urlString \(urlString)")
    }
}