Potential Problem

It does not hurt to initialize N to 3.00 in the declaration. But this might confuse a reader of the program, so it is best to not initialize N.

import java.io.*;
class  SquareRoot
{

  public static void main( String[] args ) throws IOException 
  {
    final double smallValue = 1.0E-14 ;
    double N  ;               // the user  enters a value for N
    double guess = 1.00 ;     // the same first guess works for any N

    // get the number from the user
    String NChars;
    BufferedReader stdin = 
        new BufferedReader( new InputStreamReader(System.in) );
    System.out.println("Enter the number:"); 
    NChars = stdin.readLine(); 
    N      = Double.parseDouble( NChars );

    while ( Math.abs( N/(guess*guess) - 1.0 ) > smallValue )
    {      
       guess =  N/(2*guess) + guess/2 ; // calculate a new guess
    }

    System.out.println("The square root of " + N + " is " + guess ) ;
  }

}

The program as given is largely correct, but there is a big potential problem that should be fixed.