What is wrong with this linear search?

A linear search simply means that you look through a collection (such as an array) until you find the item that you're looking for.

If the item is found you can return its index in the collection. If it's not found you can return some special number (-1 is usually used, as indices start at 0) to the calling code.

Most of your program is fine. I've filled in some code for the linear_search method and also added a few lines to obtain the number to search for in the array:

using System;
using System.Linq;

class Test
{
      static void Main()
      {
            Console.WriteLine("Press enter to see question 7");
            Console.ReadKey();
            int MaxValue;
            int RandomNumber;
            Random MyRandom;
            MaxValue = 20;
            MyRandom = new Random();
            RandomNumber = MyRandom.Next(MaxValue);

            int[] firstArray = new int[10];
            for (int i = 0; i < 10; i++)
            {
                RandomNumber = MyRandom.Next(MaxValue);
                firstArray[i] = RandomNumber;
            }
            for (int i = 0; i < 10; i++)
            {
                Console.WriteLine("Item {0} in the array is {1}", i, firstArray[i]);
            }

            //8
            Console.WriteLine("Press enter to see question 8");
            Console.ReadKey();
            Console.WriteLine("The lowest number in the array is {0}", firstArray.Min());


            //9
            Console.WriteLine("Press enter to see question 9");
            Console.ReadKey();
            int j;
            Console.Write("Enter number to search for : ");
            j = int.Parse(Console.ReadLine());
            int index = linear_search(j, firstArray);            
            if (index > 0)
            {
                Console.WriteLine("{0} is found at index {1} in the array", j, index);
            }
            else
            {
                Console.WriteLine("{0} is not found in the array", j);
            }

            Console.WriteLine("Press any key to end program");
            Console.ReadKey();
      }

      static int linear_search(int search, int[] arr)
      {
            int index = -1;

            for(int i = 0; i < arr.Length; i++)
            {
               if(arr[i] == search)
               {
                  index = i;
                  break;
               }
            }
           
            return index;
      }
}