What's the difference between the "data" and "type" keywords?

type declares a type synonym. A type synonym is a new name for an existing type. For example, this is how String is defined in the standard library:

type String = [Char]

String is another name for a list of Chars. GHC will replace all usages of String in your program with [Char] at compile-time.

To be clear, a String literally is a list of Chars. It's just an alias. You can use all the standard list functions on String values:

-- length :: [a] -> Int
ghci> length "haskell"
7
-- reverse :: [a] -> [a]
ghci> reverse "functional"
"lanoitcnuf"

data declares a new data type, which, unlike a type synonym, is different from any other type. Data types have a number of constructors defining the possible cases of your type. For example, this is how Bool is defined in the standard library:

data Bool = False | True

A Bool value can be either True or False. Data types support pattern matching, allowing you to perform a runtime case-analysis on a value of a data type.

yesno :: Bool -> String
yesno True = "yes"
yesno False = "no"

data types can have multiple constructors (as with Bool), can be parameterised by other types, can contain other types inside them, and can recursively refer to themselves. Here's a model of exceptions which demonstrates this; an Error a contains an error message of type a, and possibly the error which caused it.

data Error a = Error { value :: a, cause :: Maybe (Error a) }
type ErrorWithMessage = Error String

myError1, myError2 :: ErrorWithMessage
myError1 = Error "woops" Nothing
myError2 = Error "myError1 was thrown" (Just myError1)

It's important to realise that data declares a new type which is apart from any other type in the system. If String had been declared as a data type containing a list of Chars (rather than a type synonym), you wouldn't be able to use any list functions on it.

data String = MkString [Char]
myString = MkString ['h', 'e', 'l', 'l', 'o']
myReversedString = reverse myString  -- type error

There's one more variety of type declaration: newtype. This works rather like a data declaration - it introduces a new data type separate from any other type, and can be pattern matched - except you are restricted to a single constructor with a single field. In other words, a newtype is a data type which wraps up an existing type.

The important difference is the cost of a newtype: the compiler promises that a newtype is represented in the same way as the type it wraps. There's no runtime cost to packing or unpacking a newtype. This makes newtypes useful for making administrative (rather than structural) distinctions between values.

newtypes interact well with type classes. For example, consider Monoid, the class of types with a way to combine elements (mappend) and a special 'empty' element (mempty). Int can be made into a Monoid in many ways, including addition with 0 and multiplication with 1. How can we choose which one to use for a possible Monoid instance of Int? It's better not to express a preference, and use newtypes to enable either usage with no runtime cost. Paraphrasing the standard library:

-- introduce a type Sum with a constructor Sum which wraps an Int, and an extractor getSum which gives you back the Int
newtype Sum = Sum { getSum :: Int }
instance Monoid Sum where
    (Sum x) `mappend` (Sum y) = Sum (x + y)
    mempty = Sum 0

newtype Product = Product { getProduct :: Int }
instance Monoid Product where
    (Product x) `mappend` (Product y) = Product (x * y)
    mempty = Product 1

type works just like let: it allows you to give a re-usable name to something, but that something will always work just as if you had inlined the definition. So

type ℝ = Double

f :: ℝ -> ℝ -> ℝ
f x y = let x2 = x^2
        in x2 + y

behaves exactly the same way as

f' :: Double -> Double -> Double
f' x y = x^2 + y

as in: you can anywhere in your code replace f with f' and vice versa; nothing would change.

OTOH, both data and newtype create an opaque abstraction. They are more like a class constructor in OO: even though some value is implemented simply in terms of a single number, it doesn't necessarily behave like such a number. For instance,

newtype Logscaledℝ = LogScaledℝ { getLogscaled :: Double }

instance Num LogScaledℝ where
  LogScaledℝ a + LogScaledℝ b = LogScaledℝ $ a*b
  LogScaledℝ a - LogScaledℝ b = LogScaledℝ $ a/b
  LogScaledℝ a * LogScaledℝ b = LogScaledℝ $ a**b

Here, although Logscaledℝ is data-wise still just a Double number, it clearly behaves different from Double.

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