expected constructor, destructor, or type conversion before ‘(’ token

The first constructor in the header should not end with a semicolon. #include <string> is missing in the header. string is not qualified with std:: in the .cpp file. Those are all simple syntax errors. More importantly: you are not using references, when you should. Also the way you use the ifstream is broken. I suggest learning C++ before trying to use it.

Let’s fix this up:

//polygone.h
# if !defined(__POLYGONE_H__)
# define __POLYGONE_H__

#include <iostream>
#include <string>    

class Polygone {
public:
  // declarations have to end with a semicolon, definitions do not
  Polygone(){} // why would we needs this?
  Polygone(const std::string& fichier);
};

# endif

and

//polygone.cc
// no need to include things twice
#include "polygone.h"
#include <fstream>


Polygone::Polygone(const std::string& nom)
{
  std::ifstream fichier (nom, ios::in);


  if (fichier.is_open())
  {
    // keep the scope as tiny as possible
    std::string line;
    // getline returns the stream and streams convert to booleans
    while ( std::getline(fichier, line) )
    {
      std::cout << line << std::endl;
    }
  }
  else
  {
    std::cerr << "Erreur a l'ouverture du fichier" << std::endl;
  }
}