C# error "Type 'string' does not contain a definition for 'equals' and no extension method 'equals' of type 'string' could be found."

You are not missing any namespace in your C# code, you are already using System for .Equals, "E" just need to be in capital letter 

Here is the code which worked for me when executed in my local Visual studio

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;

namespace TimeConversion
{
    class Program
    {
        /*
   * Complete the timeConversion function below.
   */
        static string timeConversion(string s)
        {
            /*
             * Write your code here.
             */
            String result = string.Empty;
            String strconv = s.Substring(0, 2);
            int convt = Int32.Parse(strconv);
            if (s.Substring(8).Equals("AM"))
            {
                if (s.Substring(0, 2).Equals("12"))
                {
                    result = s.Substring(0, 8);
                }
            }
            else if (s.Substring(8).Equals("PM"))
            {
                if (s.Substring(0, 2).Equals("24") || convt == 12)
                {
                    result = "00" + s.Substring(0, 2);
                }
                else if (convt < 12)
                {
                    convt = convt + 12;
                    result = convt + s.Substring(2, 8);
                }
            }
            return result;
        }

        static void Main(string[] args)
        {
            TextWriter tw = new StreamWriter(@"E:\Outputtime.txt", true);
            string s = Console.ReadLine();
            string result = timeConversion(s);
            tw.WriteLine(result);
            tw.Flush();
            tw.Close();
        }
    }
}

Here is the screenshot which shows Code/Output

As you can see in the above code you need to replace ".equals()" to ".Equals()".

Thank you so much   pika :) .