How do I initialize a list of object in an array of object

After this line:

Author[] author = new Author[5];

all elements of author are null and not an actual instance of the Author class. So the first thing to do when entering the first loop, is creating an instance of Author and putting it in the array.

Another problem is that, when creating an Author instance, the book member is also null. When an instance of Author is created, you have to assign a value to the book property. You can do this using a constructor: Constructors (C# Programming Guide)[^]

Another problem is that you try to access an element of author[i].book, but that element does not exist yet at index j. You first have to add it.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
 
namespace Test
{
    class Class6
    {
        static void Main (string[] args)
        {
            Author[] author = new Author[5];
            int bookCount = 3;
            for (int i = 0; i < author.Length; i++ )
            {
                author[i] = new Author();
                for (int j = 0; j < bookCount; j++)
                {
                    author[i].book.Add(new Book());
                    author[i].book[j].Title = "book title";
                    author[i].book[j].Language = "book language";
                    // or, as a one-liner: author[i].book.Add(new Book() { Title = "book title", Language = "book language" });
                }
            }
 
            Console.Read();
        }
 
    }
 
    public class Author
    {
        public List<Book> book {get; set;}

        public Author()
        {
            book = new List<Book>();
        }
    }
 
    public class Book
    {
        public string Title {get; set;}
        public string Language {get;set;}
    }
}

I don't like working with arrays when setting new items. The Framework has the built in List<t> construct which makes doing things like this much easier. You also don't have to worry about the explicit size of your array.

List<author> authors = new List<author>();
int authorCount = 5;
int bookCount = 3;
for(int i = 0; i < authorCount; i++)
{
    var newAuthor = new Author();
    for (int j = 0; j < bookCount; j++)
    {
        newAuthor.book.Add(new Book{
            Title = "book title";
            Language = "book language";
        });
    }
    authors.Add(newAuthor);
}

To access the authors list as an array all you need to call is authors.ToArray();