A copy constructor is used to initialize a previously uninitialized object from some other object's data.
A(const A& rhs) : data_(rhs.data_) {}
For example:
A aa;
A a = aa; //copy constructor
An assignment operator is used to replace the data of a previously initialized object with some other object's data.
A& operator=(const A& rhs) {data_ = rhs.data_; return *this;}
For example:
A aa;
A a;
a = aa; // assignment operator
You could replace copy construction by default construction plus assignment, but that would be less efficient.
(As a side note: My implementations above are exactly the ones the compiler grants you for free, so it would not make much sense to implement them manually. If you have one of these two, it's likely that you are manually managing some resource. In that case, per The Rule of Three, you'll very likely also need the other one plus a destructor.)