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);
}

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