Your Syntax is not correct.
Suppose For Example , I want to make a Function That takes a number and returns it square , I would go like ,
#include<stdio.h>
int square(int);    //Function Prototype Declaration Telling Compiler  
                    //That there would be function named square whose        
                    //takes an integer and whose return type is integer.
int main()
{
   int x = 4;
   int y ;
   y = square(x);          //Calling Square Function passing Value `x`
                           //And Storing it in `y`.
   printf("%d\n" , y);     //Printing `y`
   //Or You Could As Well Do This
   printf("%d" , square(x));      //Without Storing Printing return
                                  //value of square(x)
  return 0;
 }
 int square(int k){   //Function Definition Contains The Code What The      
  int r = k*k;        //Function Does , Here it take the value passed      
  return r;           //to the function in this case `x` and store it 
                      //in `k` , and then initialise `r` to be `k*k` 
  }                   //and then returns `  
So in your code 
  int yrBorn(int);
  #include<stdio.h>
  int yrBorn (int yearBorn)    
  {   
      printf("Enter the year you were born in: ");
      scanf("%d", &yearBorn);
      return yearBorn;
  }
   int main ()
   {       
     printf("%d",yrBorn);      //Here You are not calling function.
     printf("%d" , yrBorn(0)); //This will work as in your prototype
                               //declaration you have declared
                               //function to take a int parameter and 
                               //return int
   }
What You Would Wanted I Suppose Was this 
  #include<stdio.h>
  int yrBorn(void);
  int yrBorn (void)   //Here Function is Expected to take no arguments 
                      //and return int value.
  {
    int yearBorn;
    printf("Enter Your Birth Year : ");
    scanf("%d" , &yearBorn);
    return yearBorn;
   }    
 int main(){
     printf("%d" , yrBorn());     //You Would Call it like this 
   }
Notice there is nothing between parenthesis it is because function does not accept any arguments.
Please Read A Decent C Book with complete dedication forgetting everything what you already know or you would face similar misconceptions later too , I would recommend Watching this series Nptel - Introduction to C Programming and checking this The Definitive C Book Guide and List for Best C Programming Books For Beginners.