Pointer to array of base class, populate with derived class

Your array is of the wrong type: it stores BaseClass object instances instead of pointers to them. Since BaseClass seems to be abstract, the compiler complains that it cannot default-construct instances to fill your array.

Even if BaseClass were not abstract, using arrays polymorphically is a big no-no in C++ so you should do things differently in any case.

Fix this by changing the code to:

BaseClass** base = new BaseClass*[2];

base[0] = new FirstDerivedClass;
base[1] = new SecondDerivedClass;

That said, most of the time it is preferable to use std::vector instead of plain arrays and smart pointers (such as std::shared_ptr) instead of dumb pointers. Using these tools instead of manually writing code will take care of a host of issues transparently at an extremely small runtime cost.

It is C++ use std::vector instead of simple array:

std::vector<BaseClass*> base;
base.push_back(new FirstDerivedClass());
base.push_back(new SecondDerivedClass());

As Kerrek SB noticed safest method is to use std::unique_ptr:

std::vector<std::unique_ptr<BaseClass> > base;
base.push_back( std_unique_ptr<BaseClass>(new FirstDerivedClass()) );
base.push_back( std_unique_ptr<BaseClass>(new SecondDerivedClass()) );