Why's the downloadStatus a constant? Also, I don't understand the constant used under the case. success.

Why's the downloadStatus a constant?

In the context of the example- we don't need to change its value at any point. So a constant is suitable. The documentation clarifies this:

"If a stored value in your code is not going to change, always declare it as a constant with the let keyword. Use variables only for storing values that need to be able to change."

Also, I don't understand the constant used under the case. success.

All we're doing here is assigning the value of .Success to a new constant called success. I think its often confusing when the examples use the same words, but we could have just as easily used:

switch downloadStatus {
case .Success(let fred):
    println(fred)
case .Failure(let simon):
    println(simon)
}