=> has several meanings in Scala, all related to its mathematical meaning as implication.
In a value, it introduces a function literal, or lambda. e.g. the bit inside the curly braces in List(1,2,3).map { (x: Int) => x * 2 }
In a type, with symbols on both sides of the arrow (e.g. A => T, (A,B) => T, (A,B,C) => T, etc.) it's sugar for Function[A[,B,...],T], that is, a function that takes parameters of type A[,B...], and returns a value of type T.
Empty parens on the left hand side (e.g. () => T) indicate that the function takes no parameters (also sometimes called a "thunk");
Empty parens on the right hand side denote that it returns ()—the sole value of type Unit, whose name can also be written ()—confused yet? :)
A function that returns Unit is also known as a procedure, normally a method that's called only for its side effect.
In the type declaration for a method or function parameter, with no symbol on the left hand side (e.g. def f(param: => T)) it's a "by-name parameter", meaning that is evaluated every time it's used within the body of the function, and not before. Ordinary "by-value" parameters are evaluated before entry into the function/method.
In a case clause, they separate the pattern (and optional guard) from the result expression, e.g. case x => y.
