iPhone SDK: How do you download video files to the Document Directory and then play them?

Your code will work to play a movie file.

The simplest way to download is synchronously:

NSData *data = [NSData dataWithContentsOfURL:movieUrl];
[data writeToURL:movieUrl atomically:YES];

But it is better (for app responsiveness, etc) to download asynchronously:

    NSURLRequest *theRequest = [NSURLRequest requestWithURL:movieUrl cachePolicy:NSURLRequestReloadIgnoringLocalCacheData timeoutInterval:60];
    receivedData = [[NSMutableData alloc] initWithLength:0];
        NSURLConnection *connection = [[NSURLConnection alloc] initWithRequest:theRequest delegate:self startImmediately:YES];

This requires implementing the informal NSURLConnection protocol:

- (void) connection:(NSURLConnection *)connection didReceiveResponse:(NSURLResponse *)response {
    [UIApplication sharedApplication].networkActivityIndicatorVisible = YES;
    [receivedData setLength:0];
}

- (void) connection:(NSURLConnection *)connection didReceiveData:(NSData *)data {
    [receivedData appendData:data];
}

- (void) connection:(NSURLConnection *)connection didFailWithError:(NSError *)error {
    [UIApplication sharedApplication].networkActivityIndicatorVisible = NO;
    [connection release];
}

- (NSCachedURLResponse *) connection:(NSURLConnection *)connection willCacheResponse:(NSCachedURLResponse *)cachedResponse {
    return nil;
}

- (void) connectionDidFinishLoading:(NSURLConnection *)connection {
    [UIApplication sharedApplication].networkActivityIndicatorVisible = NO;
    [connection release];
    [self movieReceived];
}

and then saving (and playing) the movie file in the movieReceived method.