Nintendo has been in this game for quite some time, so it's no news that they included security mechanisms to prevent running unauthorised software or games from a different region. Furthermore, due to the new range of I/O that the GameCube provides, the attack surface got significantly larger.
Security mechanisms
We can organise them into these areas:
DVD controller
Even though this is the first Nintendo console to use the disc medium, attempting to play pirated copies of games just wasn't going to be easy. The miniDVD is protected by using proprietary bar codes on the inner side of the disc, in addition to having its data encrypted. The validation and decryption process works seamlessly: The miniDVD controller takes care of it while the system is limited to only requesting the data.
The hardware composing the DVD reader can be imagined as a fortress wall which is only accessed using a series of commands, the miniDVD controller features a proprietary CPU that will take care of deciding if the inserted disc is genuine or not, and if it's not, no command issued by the main CPU will convince to read it otherwise.
Defeat: As with any other cat-and-mouse game, it was just a matter of time before third-party companies successfully reverse-engineered the controller to build mod-chips that could trick the reader. But bear in mind that no mod-chip will make this console magically fit a conventional CD/DVD without physically altering the case!
IPL and EXI
Another possible path of exploitation consists of using the external I/O available to load Homebrew programs. Although, without cracking the DVD reader first, the only other option available is to try to take control of the first program that the GameCube loads, and that is... The IPL.
That means that by reversing engineering the BIOS and replacing the chip with a modified one, one would be able to run, let's say, a file reader, and from there execute programs received from the accessory ports (assuming the additional hardware is plugged in).
Be as it may, the IPL chip is encrypted using XOR conditionals and a Cipher-text , making it 'impossible' to reverse engineer.
Defeat: Hackers eventually discovered that the hardware that handles the decryption of the IPL contained a bug that enabled them to capture the Cipher-text used . With this, another ROM could be constructed and encrypted with the same cypher so the GameCube boots it as its own!
As if that wasn't enough, hackers also found new methods to trick the miniDVD reader into loading conventional DVDs.
Honourable Mention
Before those two mechanisms were discovered, there was actually a much simpler way of loading arbitrary code without any modification whatsoever. This method consisted of hijacking the online protocol.
Some games like Phantasy Star Online implemented their own online functionality, which could be updated by downloading an updated executable (DOL file) from the company's servers, and the latter didn't implement any security in its protocol. So, as you can see, this was a man-in-the-middle attack waiting to happen...
Long story short, by spoofing a fake server the GameCube would just download (and execute) whatever DOL you could provide. That means hackers only needed the original game and the broadband adapter. This technique is known as PSOload .