Either declare a explicit constructor at your subclass that throws FileNotFoundException:

public MySubClass() throws FileNotFoundException {
} 

Or surround the code in your base class with a try-catch block instead of throwing a FileNotFoundException exception:

public MyBaseClass()  {
    FileInputStream fstream = null;
    try {
        File inFile = new File("textfile.txt");
        fstream = new FileInputStream(inFile);
        // Get the object of DataInputStream
        DataInputStream in = new DataInputStream(fstream);
        BufferedReader br = new BufferedReader(new InputStreamReader(in));
        // Do something with the stream
    } catch (FileNotFoundException ex) {
        Logger.getLogger(this.getClass().getName()).log(Level.SEVERE, null, ex);
    } finally {
        try {
            // If you don't need the stream open after the constructor
            // else, remove that block but don't forget to close the 
            // stream after you are done with it
            fstream.close();
        } catch (IOException ex) {
            Logger.getLogger(this.getClass().getName()).log(Level.SEVERE, null, ex);
        }
    }  
} 

Unrelated, but since you are coding a Java applet, remember that you will need to sign it in order to perform IO operations.

You need to surround your code with try and catch as follows:

try {
    File inFile = new File("textfile.txt");
    FileInputStream fstream = new FileInputStream(inFile);//ERROR
} catch (FileNotFoundException fe){
    fe.printStackTrace();
}
// Get the object of DataInputStream
DataInputStream in = new DataInputStream(fstream);
BufferedReader br = new BufferedReader(new InputStreamReader(in));

This is guesswork as we don't have the complete code.

From the Javadoc: public FileInputStream(File file) throws FileNotFoundException

It means that when you do a new FileInputStream() like you do, it can come back with a FileNotFoundException. This is a checked exception, that you need to either rethrow (i.e. add 'throws FileNotFoundException' in the method where you do the new) or catch (see other try/catch responses).

I would say that the textfile.txt does not exist or is not in the current directory.

So you are trying to create the FileInputStream on a file that does not exist.

try doing a

System.out.println(inFile.getAbsolutePath() + " " + inFile.exists());

on the file and make sure that it does exist before you try and use it.