My first attempt at Racket.

map.rkt

; List comprehension version, more Racket-esque, I'm told.
(define (mymap lam l) ; lam is a lambda (see bottom) and l a list
  (for/list ([i l])   ; i is given element for the iteration, l the list
            (lam i))) ; The expression to be evaluated on each iteration

; From scratch (kind of). Recursively creates a list of the results of calling
; a lambda on each element
(define (mymap lam l)               ; Define a function that takes a lambda and a list.
  (if (empty? l)                    ; Base case, if the list is empty:
      empty                         ;   return an empty list
      (cons (lam (first l))         ;   else, build a list where the first element is the
            (mymap lam (rest l))))) ;     result of running lambda on the first element
                                    ;     and the rest is the result of running 'mymap'
                                    ;     on the remaining elements.

; Call mymap, passing in a lambda that will be called on each element.
(mymap (lambda (e)              ; Pass in a lambda as the first argument.
         (string-append e "!")) ; Body of lambda.
       (list "one" "two"))      ; Second argument of mymap, a list.

; Output:
; ("one!" "two!")