The current problem is that no field has the name "address.description" (it's not even a valid name for a field).

You have to get the address field first, access its class, and get the nested field from there. To do so, use getType():

Agreement agreement = new Agreement();
Class c = agreement.getClass();          // Agreement class
Field f = c.getDeclaredField("address"); // address field
Class<?> fieldClass = f.getType();       // class of address field (Address)
Field nested = fieldClass.getDeclaredField("description"); // description field

You can also chain the calls without local variables:

Field nested = c.getDeclaredField("address").getType().getDeclaredField("description");

The method Class.getDeclaredFields will get you an array of Fields representing each field of the class. You could loop over these and check the type returned by Field.getType. Filtering fields of type List to List<String> is trickier - see this post for help with that.

After doing the first dynamic lookup of the fields you want, you should keep track of (memoize) the relevant Field objects for better performance.

Here's a quick example:

//for each field declared in User,
for (Field field : User.class.getDeclaredFields()) {
    //get the static type of the field
    Class<?> fieldType = field.getType();
    //if it's String,
    if (fieldType == String.class) {
        // save/use field
    }
    //if it's String[],
    else if (fieldType == String[].class) {
        // save/use field
    }
    //if it's List or a subtype of List,
    else if (List.class.isAssignableFrom(fieldType)) {
        //get the type as generic
        ParameterizedType fieldGenericType =
                (ParameterizedType)field.getGenericType();
        //get it's first type parameter
        Class<?> fieldTypeParameterType =
                (Class<?>)fieldGenericType.getActualTypeArguments()[0];
        //if the type parameter is String,
        if (fieldTypeParameterType == String.class) {
            // save/use field
        }
    }
}

Note that I used reference equality (==) instead of isAssignableFrom to match String.class and String[].class since String is final.

Edit: Just noticed your bit about finding nested UserDefinedObjects. You could apply recursion to the above strategy in order to search for those.

You should pass the object to get method of the field, so

  import java.lang.reflect.Field;

  Field field = object.getClass().getDeclaredField(fieldName);    
  field.setAccessible(true);
  Object value = field.get(object);

 ClassA classA=new ClassA();
       Field[] fields =classA.getClass().getDeclaredFields();
       for (Field field : fields) {
           if (Collection.class.isAssignableFrom(field.getType())){
               System.out.println(field.getName());//you can remove this as per requirement
               Class<?> c1 = (Class<?>) ((((ParameterizedType) field.getGenericType()).getActualTypeArguments())[0]);

                Field[] fieldsInner = c1.getDeclaredFields();
                for (Field fieldInner : fieldsInner) {
                    System.out.println(fieldInner.getName());

           }
           else{
           System.out.println(field.getName());}

       }