A double-pointer, is simply a pointer to a pointer. A common usage is for arrays of character strings. Imagine the first function in just about every C/C++ program:
int main(int argc, char *argv[])
{
...
}
Which can also be written
int main(int argc, char **argv)
{
...
}
The variable argv is a pointer to an array of pointers to char. This is a standard way of passing around arrays of C "strings". Why do that? I've seen it used for multi-language support, blocks of error strings, etc.
Don't forget that a pointer is just a number - the index of the memory "slot" inside a computer. That's it, nothing more. So a double-pointer is index of a piece of memory that just happens to hold another index to somewhere else. A mathematical join-the-dots if you like.
This is how I explained pointers to my kids:
Imagine the computer memory is a series of boxes. Each box has a number written on it, starting at zero, going up by 1, to however many bytes of memory there is. Say you have a pointer to some place in memory. This pointer is just the box number. My pointer is, say 4. I look into box #4. Inside is another number, this time it's 6. So now we look into box #6, and get the final thing we wanted. My original pointer (that said "4") was a double-pointer, because the content of its box was the index of another box, rather than being a final result.
It seems in recent times pointers themselves have become a pariah of programming. Back in the not-too-distant past, it was completely normal to pass around pointers to pointers. But with the proliferation of Java, and increasing use of pass-by-reference in C++, the fundamental understanding of pointers declined - particularly around when Java became established as a first-year computer science beginners language, over say Pascal and C.
I think a lot of the venom about pointers is because people just don't ever understand them properly. Things people don't understand get derided. So they became "too hard" and "too dangerous". I guess with even supposedly learned people advocating Smart Pointers, etc. these ideas are to be expected. But in reality there a very powerful programming tool. Honestly, pointers are the magic of programming, and after-all, they're just a number.