Every time I try to start the game, I get a black window and a dialog box called "Giants Error" with two options, "Ok" or "Cancel".

Any idea what this means/how I can fix this?

Need more details. What OS are you running this on? Have you done anything like patch it (which you shouldn't do) or add any mods (which you probably should do - see the forum sticky)? What are your hardware specs?

I have the same problem with pure GOG version 1.4 on my ancient one-core CPU machine with GeForce4 MX440 (Windows XP).

But, luckily, patch 1.497 BETA from http://www.giantswd.org/?file=2 seems to solve this problem. To be honest, I've installed patch right after GiantsHook which didn't solve the problem (I don't know if having both is a good idea; it's a fresh install with only these two patches).

The error window looked exactly like one asking to put CD into a drive and appeared at the same moment, but missed the message/text - an empty error window with two buttons and titled "Giants Error" (I know, because when I was trying to fix it, I "reapplied" patch 1.4 after which error window started asking for CD).