How can I use enum in C# for storing string constants?

Enum constants can only be of ordinal types (int by default), so you can't have string constants in enums.

When I want something like a "string-based enum" I create a class to hold the constants like you did, except I make it a static class to prevent both unwanted instantiation and unwanted subclassing.

But if you don't want to use string as the type in method signatures and you prefer a safer, more restrictive type (like Operation), you can use the safe enum pattern:

public sealed class Operation
{
    public static readonly Operation Name1 = new Operation("Name1");
    public static readonly Operation Name2 = new Operation("Name2");

    private Operation(string value)
    {
        Value = value;
    }

    public string Value { get; private set; }
}

You could do this using DescriptionAttribute, but then you'd have to write code to get the string out of the attribute.

public enum YourEnum
{
    [Description("YourName1")]
    Name1,

    [Description("YourName2")]
    Name2
}

The whole point of enums is to be ordinal constants.
However, you can achieve what you want by using an extension method:

  enum Operation
  {
      name1,
      name2
  }

  static class OperationTextExtender
  {
        public static String AsText(this Operation operation)
        {
              switch(operation)
              {
                    case Operation.name1: return "hmmm";
                    case Operation.name2: return "bdidwe";
                    ...
              }
        }
  }

  ...
  var test1 = Operation.name1;
  var test2 = test1.AsText();