Assuming the types of a and b are reference types:
In Java, == will always compare for identity - i.e. whether the two values are references to the same object. This is also called reference equality. Java doesn't have any user-defined operator overloading.
In C# it depends. Unless there's an overloaded operator which handles it, == will behave like Java (i.e. comparing for reference equality). However, if there's an overload which matches the compile-time types of a and b (e.g. if they're both declared as strings) then that overload will be called. That can behave how it wants, but it typically implements value equality (i.e. a and b can refer to different but equal values and it would still return true).
In both languages, a.Equals(b) or a.equals(b) will call the virtual Equals/equals method declared by Object, unless a more specific overload has been introduced by the compile-time type of a. This may or may not be overridden in the execution-time type of the object that a refers to. In both .NET and Java, the implementation in Object also checks for identity. Note that this depends on the execution-time type rather than the compilation-time type that overload resolution depends on.
Of course, if a is null then you'll get a NullReferenceException/NullPointerException when you try to call a.equals(b) or a.Equals(b).
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