How does overloading of const and non-const functions work?

The compiler's "algorithm" is like this: Every member function of class X has an implicit argument of type X& (I know, most think it's X*, but the standard states, that for purposes of Overload Resolution we assume it to be a reference). For const functions, the type of the argument is const X&. Thus the algorithm, if a member function is called the two versions, const and non-const, are both viable candidates, and the best match is selected just as in other cases of overload resolution. No magic :)

In the example you gave:

vector<int>::const_iterator it = myvector.begin();

if myvector isn't const the non-const version of begin() will be called and you will be relying on an implicit conversion from iterator to const_iterator.