Why don't std::vector's elements need a default constructor?
And how can I write my own array class to not need a default constructor for its elements? Right now, when I do the new [] to allocate space, I need a default constructor.
std::vector does not.
How do they do this magic?
Answer
std::vector doesn't need the default constructor because it never uses it. Every time it needs to construct an element, it does it by using the copy constructor, because every time it has something to copy: either existing vector element or an element you yourself supplied for copying through a method's parameter (explicitly or implicitly, by relying on a default argument)
You can write a class like that in exactly the same way: every time you need to construct a new element in your array, require the user to supply an element for copying. In this case constructing that original element becomes user's responsibility.
Every time it appears as if std::vector "requires" a default constructor from you, it simply means that somewhere you relied on a default argument of some of the vectors methods, i.e. it was you who tried to default-construct an element, not the vector. The vector itself, again, will never try to default-construct elements.
In order to avoid the default constructor requirement during memory allocation, standard library allocates raw uninitialized memory block and then immediately copy-constructs new elements in that raw memory block (which is something new[] cannot do). This functionality is incapsulated in std::allocator class. You can use std::allocator in your code as well, meaning that the "magic" is immediately available to you too.
Note: The above applies to pre-C++11 version of C++ language specification. C++11 changed a lot of things. And these changes do create situations in which std::vector can use default constructors internally.
Also it might be worth noting that even the original C++98 specification allowed implementations to use function overloading instead of default arguments in order to implement the standard library interface. This means that formally it is possible to have a valid C++98 implementation of std::vector that uses default constructors internally.