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Let’s talk about types and memory!
In Swift a type can either be value type or reference type.

Value types

A Value type is a type that provides value semantics. But what is value semantics?
Value semantics is the behaviour you have when you manipulate Integer.

var x: Int = 0
var y: Int = x
y += 1
print(x, y) // prints 0 1

The modification of the variable y affects only the value stored into the variable y. In Swift structs and enums are value types, as you can see from the following example.

struct MyValueType { 
    var value: Int
}

var x = MyValueType(value: 0)
var y: MyValueType = x
y.value += 1
print(x.value, y.value) // prints 0 1

Reference types

You probably saw it coming: reference types are types that provide reference semantics.
In Swift reference types are classes.

class MyReferenceType { 
    var value: Int
}

var x = MyReferenceType(value: 0)
var y: MyReferenceType = x
y.value += 1
print(x.value, y.value) // prints 1 1

The modification applied to the variable y will reflect on the variable x because they both rely on shared memory.

When should you use one over the other?

Reference types are classes which means they are stored into the heap. When using them you are manipulating a pointer to the actual data, use them when you want to create a shared mutable state.
Value types are the actual data and not a reference to the data. Because of that, copies of the same value have independent states. This also means they are safer to use when running code on multiple threads (no concurrent access is possible).
When creating a value type variable it will be stored inside the stack.

Are there types that are neither value nor reference type

The notion of value/reference type is actually a bit weird, is the following struct a value type?

class SomeClass { 
    var value: Int = 0
}

struct SomeWeirdStruct {
    private let someClass = SomeClass()

    var someOtherValue: Int = 0

    var value: Int {
        get { someClass.value }
        set { someClass.value = newValue }
    }
}

var x = SomeWeirdStruct()
var y: SomeWeirdStruct = x
y.value += 1
y.someOtherValue += 1
print(x.value, y.value) // prints 1 1
print(x.someOtherValue, y.someOtherValue) // prints 0 1

You might argue that modifying y.value ends up in a modification of x.value and there is a shared mutable state between the two, yet modifying y.someOtherValue does not impact x.someOtherValue

But I think a better way to see it is that SomeWeirdStruct is a value type, but its value is a SomeClass pointer and an Integer. And even though value is declared as a variable it is a computed variable which makes it a pair of functions.
In less modern language you would write it this way :

struct SomeWeirdStruct {
    private let someClass = SomeClass()

    var someOtherValue: Int = 0

    func getValue() -> Int {
        someClass.value
    }

    func setValue(_ newValue) {
        someClass.value = newValue
    }
}

So in conclusion a type really is either a value or a reference type in Swift. But keep an open mind because saying SomeWeirdStruct is a reference type when it comes to its value variable may not be a valid statement but it is an understandable one. And if we reject such a statement, saying Array is a value type would only mean it is a struct, yet it is way more interesting to know that it provides value semantics for the elements stored in the array.