Server:
public class Server {

    private static final int PORT = 9001;

    private static HashSet<String> names = new HashSet<String>();
    private static HashSet<PrintWriter> writers = new HashSet<PrintWriter>();
    public static void main(String[] args) throws Exception {

        ServerSocket listener = new ServerSocket(PORT);
        System.out.println("The chat server is running. PORT = " + listener.getLocalPort());
        try {
            while (true) {
                new Handler(listener.accept()).start();
            }
        } finally {
            listener.close();
        }
    }

    private static class Handler extends Thread {
        private String name;
        private Socket socket;
        private BufferedReader in;
        private PrintWriter out;

        public Handler(Socket socket) {
            this.socket = socket;
        }

        public void run() {
            try {

                in = new BufferedReader(new InputStreamReader(
                        socket.getInputStream()));
                out = new PrintWriter(socket.getOutputStream(), true);

                while (true) {
                    out.println("SUBMITNAME");
                    name = in.readLine();

                    if (name == null) {
                        return;
                    }
                    synchronized (names) {
                        if (!names.contains(name)) {
                            names.add(name);

                            System.out.println(name + " has joined the chat!" + socket.getInetAddress().getHostAddress());
                            out.println("NAMEACCEPTED");
                            writers.add(out);
                            for (PrintWriter writer : writers) {
                                writer.println("MESSAGE " + name + " has joined the chat!");
                            }
                            break;
                        } else {
                            writers.add(out);
                            out.println("NAMEACCEPTED");
                        }
                    }
                }


                while (true) {
                    String input = in.readLine();
                    if (input == null) {
                        return;
                    }
                    for (PrintWriter writer : writers) {
                        writer.println("MESSAGE " + name + ": " + input);
                    }

                }
            } catch (IOException e) {
                System.out.println(e);
            } finally {
                // This client is going down!  Remove its name and its print
                // writer from the sets, and close its socket.
                if (name != null) {
                    names.remove(name);
                }
                if (out != null) {
                    writers.remove(out);
                }
                System.out.println(name + " has left the chat!");
                for (PrintWriter writer : writers) {
                    writer.println("MESSAGE " + name + " has left the chat!");
                }
                try {
                    socket.close();
                } catch (IOException e) {
                }
            }
        }
    }
}

Client:

public class Client {

    BufferedReader in;
    PrintWriter out;
    JFrame frame = new JFrame("Chatter");
    JTextField textField = new JTextField(40);
    JTextArea messageArea = new JTextArea(8, 40);

    public Client() {

        // Layout GUI
        textField.setEditable(false);
        messageArea.setEditable(false);
        frame.getContentPane().add(textField, "North");
        frame.getContentPane().add(new JScrollPane(messageArea), "Center");
        frame.pack();

        // Add Listeners
        textField.addActionListener(new ActionListener() {

            public void actionPerformed(ActionEvent e) {
                out.println(textField.getText());
                textField.setText("");
            }
        });
    }

    /**
     * Prompt for and return the address of the server.
     */
    private String getServerAddress() {
        return JOptionPane.showInputDialog(
            frame,
            "Enter IP Address of the Server:",
            "Welcome to the Chatter",
            JOptionPane.QUESTION_MESSAGE);
    }

    /**
     * Prompt for and return the desired screen name.
     */
    private String getName() {
        return JOptionPane.showInputDialog(
            frame,
            "Choose a screen name:",
            "Screen name selection",
            JOptionPane.PLAIN_MESSAGE);
    }

    /**
     * Connects to the server then enters the processing loop.
     */
    private void run() throws IOException {

        // Make connection and initialize streams
        String serverAddress = getServerAddress();
        Socket socket = new Socket(serverAddress, 9001);
        in = new BufferedReader(new InputStreamReader(
            socket.getInputStream()));
        out = new PrintWriter(socket.getOutputStream(), true);

        // Process all messages from server, according to the protocol.
        while (true) {
            String line = in.readLine();
            if (line.startsWith("SUBMITNAME")) {
                out.println(getName());
            } else if (line.startsWith("NAMEACCEPTED")) {
                textField.setEditable(true);
            } else if (line.startsWith("MESSAGE")) {
                messageArea.append(line.substring(8) + "\n");
            }
        }
    }

    /**
     * Runs the client as an application with a closeable frame.
     */
    public static void main(String[] args) throws Exception {
        Client client = new Client();
        client.frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        client.frame.setVisible(true);
        client.run();
    }
}