In addition to Jon Skeet's answer, I'd like to explain why most of the time when using == you actually get the answer true on different string instances with the same value:
string a = "Hell";
string b = "Hello";
a = a + "o";
Console.WriteLine(a == b);
As you can see, a and b must be different string instances, but because strings are immutable, the runtime uses so called string interning to let both a and b reference the same string in memory. The == operator for objects checks reference, and since both a and b reference the same instance, the result is true. When you change either one of them, a new string instance is created, which is why string interning is possible.
By the way, Jon Skeet's answer is not complete. Indeed, x == y is false but that is only because he is comparing objects and objects compare by reference. If you'd write (string)x == (string)y, it will return true again. So strings have their ==-operator overloaded, which calls String.Equals underneath.
