Eg for a definition of reverse:

reverse' :: [a] -> [a]

reverse' [] = []

reverse' (x:xs) = reverse' xs ++ [x]

In the last line of the definition, x is an element in the list (the first element) and xs represents the remainder of the list.

so if list was [1,2,3] then x is 1 and xs is [2,3]

Why are the brackets required?  And what do they signify?

Eg reverse' x:xs = reverse' xs ++ [x] results in a parse error.