vector<int>::size_type in C++

size_type is a (static) member type of the type vector<int>. Usually, it is a typedef for std::size_t, which itself is usually a typedef for unsigned int or unsigned long long.

I would read it as "declare x as a variable of a type suitable for holding the size of a vector". The vector defines its own type for its length, and it's always cleanest to use that if possible, rather than "guessing" and using int, unsigned int, long, unsigned long or size_t etc directly as you'd otherwise need to do.

vector is a template

so the vector type templated with int has a member typedef called size_type. x is defined as a variable of that type.