#include <stdio.h>

int main()
{
  /* Declare variable, Assign a initial value x. */
  char a = 'x';

  while(a != '\n')
  {
    /* Get a user input */
    a = getchar();

    if(a == '\n')
    {
      /* If the input is a new line, say good bye to user */
      printf("\nThat's it for today.\n");
    }
    else
    {
      /* Prints a user input */
      printf("%c", a);
    }
  }

  return 1;

1) a has a lousy name!
2) a has the wrong type - it should be int.

Whilst I wouldn't want to suggest that your program is otherwise devoid
of problems, it does appear to do what is asked of it, except in cases
where it hits the end-of-file, where it does get a bit messy, but you
presumably aren't worried about that.

For the record, here's how I'd have written it:

#include <stdio.h>

int main(void)
{
  int ch = 0;
  while((ch = getchar()) != EOF && ch != '\n')
  {
    putchar(ch);
  }

  printf("\nThat's it for today.\n");

  return 0;