Why do I get “Exception; must be caught or declared to be thrown” when I try to compile my Java code?

All your problems derive from this

byte[] encrypted = cipher.doFinal(toEncrypt.getBytes());
return encrypted;

Which are enclosed in a try, catch block, the problem is that in case the program found an exception you are not returning anything. Put it like this (modify it as your program logic stands):

public static byte[] encrypt(String toEncrypt) throws Exception{
    try{
        String plaintext = toEncrypt;
        String key = "01234567890abcde";
        String iv = "fedcba9876543210";

        SecretKeySpec keyspec = new SecretKeySpec(key.getBytes(), "AES");
        IvParameterSpec ivspec = new IvParameterSpec(iv.getBytes());

        Cipher cipher = Cipher.getInstance("AES/CBC/NoPadding");
        cipher.init(Cipher.ENCRYPT_MODE,keyspec,ivspec);
        byte[] encrypted = cipher.doFinal(toEncrypt.getBytes());

        return encrypted;
    } catch(Exception e){
        return null;            // Always must return something
    }
}

For the second one you must catch the Exception from the encrypt method call, like this (also modify it as your program logic stands):

public void actionPerformed(ActionEvent e)
  .
  .
  .
    try {
        byte[] encrypted = encrypt(concatURL);
        String encryptedString = bytesToHex(encrypted);
        content.removeAll();
        content.add(new JLabel("Concatenated User Input -->" + concatURL));

        content.add(encryptedTextField);
    setContentPane(content);
    } catch (Exception exc) {
        // TODO: handle exception
    }
}

The lessons you must learn from this:

• A method with a return-type must always return an object of that type, I mean in all possible scenarios
• All checked exceptions must always be handled