What does "from any type 'a' to any type 'b'" mean? That sounds like a range of types, but I don't see how that makes sense. Did you just mean that the map converts from 'a' to 'b'?
a and b are placeholders (type variables). They can be any type, for example Integer, or a Float. It doesn't matter.
[a] is a list of a, [b] is a list of b. Map takes a function that takes an a and returns a b, and applies it to every element of [a] in order to produce the [b].
An example usage is
map lessThanThree [1,2,3,4]
In this case, map takes a function where a is an integer, and b is a boolean, then takes a list of integers and produces a list of booleans.
It's also illuminating to note that haskell curries function application. This allows us to think of map as a function of one argument.
map :: (a -> b) -> ([a] -> [b])
so instead of thinking that map takes a function and a list and applies that function to the list, we can think that map takes a function and "lifts" that functions argument and result types to the list type.
This thinking leads us directly to Functors and fmap
[a] is a list of a, [b] is a list of b. Map takes a function that takes an a and returns a b, and applies it to every element of [a] in order to produce the [b]. An example usage is
In this case, map takes a function where a is an integer, and b is a boolean, then takes a list of integers and produces a list of booleans.The result is [True,True,False,False]